Properties of Right Triangles - PSAT Math

1. State the side-length ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$.

   Long leg $x\sqrt{3}$; hypotenuse $2x$. Special triangle with sides in ratio $1:\sqrt{3}:2$.

2. What is the relationship between the two acute angles in any right triangle?

   They are complementary: $\theta_1+\theta_2=90^\circ$. The two non-right angles must sum to $90°$ since triangle angles sum to $180°$.

3. Find the short leg of a $30^\circ-60^\circ-90^\circ$ triangle if the hypotenuse is $18$.

   $9$. Short leg = hypotenuse ÷ 2 in 30-60-90 triangle: $18÷2=9$.

4. Find the distance between points $(1,2)$ and $(5,5)$ using a right-triangle interpretation.

   $5$. Distance = $\sqrt{(5-1)^2+(5-2)^2}=\sqrt{16+9}=\sqrt{25}=5$.

5. In a $30^\circ-60^\circ-90^\circ$ triangle, what is the hypotenuse if the short leg is $x$?

   $2x$. Hypotenuse is twice the short leg (opposite 30°) in 30-60-90 triangles.

6. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the hypotenuse if the short leg is $9$?

   $18$. Double the short leg for 30-60-90 hypotenuse: $2 × 9 = 18$.

7. Find the short leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle with hypotenuse $18$.

   $9$. Short leg is half the hypotenuse in a 30-60-90 triangle.

8. Find the distance between points $(1,2)$ and $(7,10)$.

   $10$. $d=\sqrt{(7-1)^2+(10-2)^2}=\sqrt{36+64}=\sqrt{100}=10$.

9. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, if the short leg is $7$, what is the hypotenuse?

   $14$. Hypotenuse is twice the short leg in $30°$-$60°$-$90°$ triangles.

10. State the slope formula using points $(x_1,y_1)$ and $(x_2,y_2)$ (rise over run).

    $\frac{y_2-y_1}{x_2-x_1}$. Vertical change divided by horizontal change gives the slope.

11. Find the hypotenuse $c$ if the legs are $6$ and $8$.

    $10$. Using Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.

12. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, find the hypotenuse if a leg is $7$.

    $7\sqrt{2}$. Multiply leg by $\sqrt{2}$ for hypotenuse in 45-45-90 triangle.

13. Identify the acute angle measure if the other acute angle in a right triangle is $35^\circ$.

    $55^\circ$. Acute angles are complementary: $90°-35°=55°$.

14. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, if the hypotenuse is $18$, what is the short leg?

    $9$. Short leg is half the hypotenuse in $30°$-$60°$-$90°$ triangles.

15. Find the long leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $5$.

    $5\sqrt{3}$. In a $30°$-$60°$-$90°$ triangle, long leg = short leg × $\sqrt{3}$.

16. Find the missing leg: right triangle with hypotenuse $25$ and leg $7$; what is the other leg?

    $24$. Solve $7^2 + b^2 = 25^2$ to get $b = \sqrt{576} = 24$.

17. What is the definition of the legs in a right triangle?

    The two sides that form the $90^\circ$ angle. These perpendicular sides meet at the right angle.

18. Find the area of a right triangle with legs $9$ and $4$.

    $18$. Area $=\frac{1}{2}(9)(4)=\frac{36}{2}=18$.

19. State the formula for the area of a right triangle with legs $a$ and $b$.

    $A=\frac{1}{2}ab$. Half the product of the two legs (perpendicular sides).

20. Identify the hypotenuse in a right triangle in relation to the right angle.

    The side opposite the $90^\circ$ angle. The longest side in a right triangle, always opposite the right angle.

21. What is the tangent ratio in a right triangle for acute angle $\theta$?

    $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Ratio of the side opposite to the side adjacent to angle $\theta$.

22. State the special right triangle ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle.

    Side ratio $1:\sqrt{3}:2$. Shortest side opposite $30°$, longest opposite $90°$, middle opposite $60°$.

23. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, if a leg is $9$, what is the hypotenuse?

    $9\sqrt{2}$. Multiply leg by $\sqrt{2}$ in $45°$-$45°$-$90°$ triangles.

24. State the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ using a right-triangle interpretation.

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses Pythagorean theorem on the coordinate differences.

25. What is the converse of the Pythagorean Theorem for positive side lengths $a\le b<c$?

    If $a^2+b^2=c^2$, the triangle is right. Tests if a triangle is right-angled using side lengths.

26. Identify the hypotenuse in a right triangle in relation to the $90^\circ$ angle.

    The side opposite the $90^\circ$ angle. The longest side in a right triangle, opposite the right angle.

27. What is the formula for the hypotenuse $c$ in terms of legs $a$ and $b$?

    $c=\sqrt{a^2+b^2}$. Solve the Pythagorean theorem for the hypotenuse.

28. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the long leg in terms of short leg $x$?

    $x\sqrt{3}$. The side opposite $60°$ is $\sqrt{3}$ times the side opposite $30°$.

29. What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ derived from right triangles?

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses the Pythagorean theorem on the coordinate plane.

30. What is the formula for a leg $a$ in terms of hypotenuse $c$ and other leg $b$?

    $a=\sqrt{c^2-b^2}$. Rearrange $a^2+b^2=c^2$ to isolate the leg $a$.

31. What is the slope of a horizontal line, and what is the slope of a vertical line?

    Horizontal: $0$; vertical: undefined. Horizontal lines have no rise; vertical lines have no run.

32. Find the missing leg $a$ if a right triangle has hypotenuse $13$ and other leg $5$.

    $12$. Use $a^2+5^2=13^2$, so $a^2=169-25=144$, thus $a=12$.

33. Find the area of a right triangle with legs $8$ and $11$.

    $44$. Area = $\frac{1}{2} \times 8 \times 11 = 44$.

34. In a $30^\circ-60^\circ-90^\circ$ triangle, what is the long leg if the short leg is $x$?

    $x\sqrt{3}$. Long leg (opposite 60°) = short leg × $\sqrt{3}$ in 30-60-90 triangles.

35. What is the hypotenuse $c$ if the legs are $9$ and $12$?

    $15$. $9^2+12^2=81+144=225=15^2$ by Pythagorean theorem.

36. State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $x$.

    $x:x:x\sqrt{2}$. Isosceles right triangle has equal legs and hypotenuse $\sqrt{2}$ times longer.

37. In a right triangle, what is the relationship between the two acute angles?

    They are complementary: sum is $90^\circ$. The three angles sum to $180°$, with one being $90°$.

38. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, find the hypotenuse if the short leg is $9$.

    $18$. Hypotenuse is twice the short leg in 30-60-90 triangles.

39. Find the long leg in a $30^\circ$-$60^\circ$-$90^\circ$ triangle if the short leg is $4$.

    $4\sqrt{3}$. Long leg = short leg × $\sqrt{3}$ in $30°$-$60°$-$90°$ triangle.

40. Find the short leg if a $30^\circ$-$60^\circ$-$90^\circ$ triangle has hypotenuse $18$.

    $9$. Short leg = hypotenuse ÷ 2 in a 30-60-90 triangle.

41. What is the short leg if a $30^\circ$-$60^\circ$-$90^\circ$ triangle has hypotenuse $18$?

    $9$. Short leg is half the hypotenuse in $30°$-$60°$-$90°$ triangle.

42. Which inequality classifies a triangle as acute using sides $a\le b<c$?

    Acute if $a^2+b^2>c^2$. When leg squares exceed hypotenuse square, all angles are less than $90°$.

43. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is the hypotenuse if each leg is $7$?

    $7\sqrt{2}$. Apply the $45°$-$45°$-$90°$ ratio: hypotenuse = leg × $\sqrt{2}$.

44. State the definition of the altitude to the hypotenuse in a right triangle.

    A segment from the right angle perpendicular to the hypotenuse. Creates a perpendicular from the vertex angle to the opposite side.

45. Identify whether side lengths $7,24,25$ form a right triangle.

    Yes, because $7^2+24^2=25^2$. Check: $49+576=625$, which equals $25^2$.

46. What is the $30^\circ$-$60^\circ$-$90^\circ$ side ratio (short, long, hypotenuse)?

    $1:\sqrt{3}:2$. Sides opposite $30°$, $60°$, and $90°$ angles follow this ratio.

47. What is the distance between points $(1,2)$ and $(5,5)$?

    $5$. $d=\sqrt{(5-1)^2+(5-2)^2}=\sqrt{16+9}=\sqrt{25}=5$.

48. Find the missing leg $b$ if $c=13$ and $a=5$ in a right triangle.

    $12$. Use $b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.

49. State the definition of a Pythagorean triple in terms of integers $a,b,c$ with $c$ largest.

    Integers with $a^2+b^2=c^2$. Three positive integers satisfying the Pythagorean theorem.

50. State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$.

    short $x$, long $x\sqrt{3}$, hypotenuse $2x$. Special triangle with sides in ratio $1:\sqrt{3}:2$.

51. State the side ratio for a $45^\circ-45^\circ-90^\circ$ right triangle in simplest radical form.

    $1:1:\sqrt{2}$. Equal legs to hypotenuse ratio in an isosceles right triangle.

52. Find $c$ if a right triangle has legs $6$ and $8$.

    $10$. Using $6^2+8^2=36+64=100$, so $c=\sqrt{100}=10$.

53. What is the shorter leg if a $30^\circ$-$60^\circ$-$90^\circ$ triangle has hypotenuse $2x$?

    $x$. In a $30°$-$60°$-$90°$ triangle, shortest leg is half the hypotenuse.

54. What similarity relationship is created by drawing the altitude to the hypotenuse of a right triangle?

    The $2$ smaller triangles are similar to the original and to each other. All three triangles share the same angle measures.

55. Find the hypotenuse $c$ if the legs are $9$ and $12$.

    $15$. $c = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$.

56. State the side-length ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle (short:long:hypotenuse).

    $1:\sqrt{3}:2$. Sides opposite $30°$, $60°$, and $90°$ angles respectively.

57. State the Pythagorean Theorem for a right triangle with legs $a,b$ and hypotenuse $c$.

    $a^2+b^2=c^2$. Relates the squares of the legs to the square of the hypotenuse.

58. Identify the hypotenuse in a right triangle in terms of its position relative to the $90^\circ$ angle.

    The side opposite the $90^\circ$ angle. The longest side in a right triangle, always opposite the right angle.

59. State the slope formula for a line through $(x_1,y_1)$ and $(x_2,y_2)$.

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run; change in $y$ over change in $x$.

60. State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$.

    Sides $x,x\sqrt{3},2x$. Short leg : long leg : hypotenuse = $1 : \sqrt{3} : 2$.

61. Identify whether $7,24,25$ can be side lengths of a right triangle.

    Yes, since $7^2+24^2=25^2$. Check if Pythagorean theorem holds: $49+576=625$ ✓.

62. What is the area formula for a right triangle with legs $a$ and $b$ as base and height?

    $A=\frac{1}{2}ab$. Half the product of the two perpendicular sides.

63. State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle in simplest form.

    $1:1:\sqrt{2}$. Isosceles right triangles have two equal legs and hypotenuse $\sqrt{2}$ times a leg.

64. Find $c$ for a right triangle with legs $6$ and $8$.

    $10$. Apply $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.

65. What are the side lengths in the Pythagorean triple with legs $5$ and $12$?

    $5,12,13$. The hypotenuse is $\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$.

66. Identify the hypotenuse in a right triangle in relation to the right angle.

    The side opposite the $90^\circ$ angle. The longest side in a right triangle, always opposite the right angle.

67. State the Pythagorean triple obtained by scaling $3,4,5$ by a factor of $k$.

    $3k,4k,5k$. Multiplying each side by $k$ preserves the right angle.

68. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $x$, what is the hypotenuse?

    $x\sqrt{2}$. In isosceles right triangles, hypotenuse equals leg times $\sqrt{2}$.

69. State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle in simplest form.

    $1:\sqrt{3}:2$. Short leg opposite $30°$, long leg opposite $60°$, hypotenuse opposite $90°$.

70. Find the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $9$.

    $9\sqrt{2}$. In $45°$-$45°$-$90°$ triangle, hypotenuse equals leg times $\sqrt{2}$.

71. Find the hypotenuse of a $45^\circ-45^\circ-90^\circ$ triangle with leg $7$.

    $7\sqrt{2}$. Multiply leg by $\sqrt{2}$ in a 45-45-90 triangle: $7\times\sqrt{2}$.

72. State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle in simplest form.

    $1:\sqrt{3}:2$. Short leg : long leg : hypotenuse in this special triangle.

73. What is the hypotenuse if a $45^\circ$-$45^\circ$-$90^\circ$ triangle has leg $x$?

    $x\sqrt{2}$. In a $45°$-$45°$-$90°$ triangle, hypotenuse equals leg times $\sqrt{2}$.

74. Find the missing leg $b$ if $c=13$ and $a=5$ in a right triangle.

    $12$. Using $5^2+b^2=13^2$, so $b^2=169-25=144$, thus $b=12$.

75. State the side-length ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg length $x$.

    Legs $x,x$; hypotenuse $x\sqrt{2}$. Isosceles right triangle has equal legs and hypotenuse $\sqrt{2}$ times longer.

76. State the side-length ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle.

    $1:1:\sqrt{2}$. Legs are equal, hypotenuse is $\sqrt{2}$ times a leg.

77. Find the distance between $(1,2)$ and $(4,6)$.

    $5$. $d=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=\sqrt{25}=5$.

78. Find the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $x=9$.

    $9\sqrt{2}$. In 45-45-90 triangle, hypotenuse = leg × $\sqrt{2}$.

79. Find the hypotenuse if a $45^\circ$-$45^\circ$-$90^\circ$ triangle has leg $7$.

    $7\sqrt{2}$. In 45-45-90 triangle, hypotenuse = leg × $\sqrt{2}$.

80. What is the distance formula derived from the Pythagorean Theorem for points $(x_1,y_1)$ and $(x_2,y_2)$?

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses the Pythagorean theorem on the coordinate plane.

81. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the short leg if the hypotenuse is $18$?

    $9$. Short leg = hypotenuse ÷ 2 in a $30°$-$60°$-$90°$ triangle.

82. State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg length $x$.

    Sides $x,x,x\sqrt{2}$. Isosceles right triangle has legs equal, hypotenuse $\sqrt{2}$ times longer.

83. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the hypotenuse in terms of short leg $x$?

    $2x$. The hypotenuse is twice the side opposite $30°$.

84. Find the hypotenuse in a $30^\circ$-$60^\circ$-$90^\circ$ triangle if the short leg is $6$.

    $12$. Hypotenuse = $2$ × short leg in $30°$-$60°$-$90°$ triangle.

85. What is the converse of the Pythagorean Theorem for side lengths $a,b,c$ with $c$ largest?

    If $a^2+b^2=c^2$, the triangle is right. If sides satisfy the Pythagorean equation, then the angle is $90°$.

86. Find the distance between $(2,3)$ and $(8,11)$.

    $10$. Distance formula: $\sqrt{(8-2)^2+(11-3)^2}=\sqrt{36+64}=\sqrt{100}$.

87. Find $c$ if a right triangle has legs $9$ and $12$.

    $15$. Apply Pythagorean theorem: $c=\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$.

88. State the side ratios for a $30^\circ$-$60^\circ$-$90^\circ$ triangle (short, long, hypotenuse).

    $x,\ x\sqrt{3},\ 2x$. Special triangle with sides in ratio $1:\sqrt{3}:2$.

89. State the Pythagorean identity relating $\sin(\theta)$ and $\cos(\theta)$.

    $\sin^2(\theta)+\cos^2(\theta)=1$. Follows from dividing the Pythagorean theorem by the hypotenuse squared.

90. State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$.

    Sides $x,x\sqrt{3},2x$. Opposite $30°$, $60°$, $90°$ angles respectively.

91. State the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$ using a right triangle.

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Derived from the Pythagorean theorem applied to coordinate differences.

92. State the distance formula that comes from the Pythagorean Theorem for points $(x_1,y_1)$ and $(x_2,y_2)$.

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Pythagorean theorem applied to coordinate differences.

93. What is the missing leg $b$ if the hypotenuse is $13$ and the other leg is $5$?

    $12$. $b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.

94. Find the area of a right triangle with legs $8$ and $15$.

    $60$. Area = $\frac{1}{2} \times 8 \times 15 = 60$.

95. In a right triangle, if the legs are $6$ and $8$, what is the hypotenuse length?

    $10$. Apply Pythagorean theorem: $6^2+8^2=36+64=100=10^2$.

96. State the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$ using a right triangle.

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses Pythagorean theorem on coordinate differences.

97. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is the hypotenuse in terms of leg $x$?

    $x\sqrt{2}$. In isosceles right triangles, hypotenuse equals leg times $\sqrt{2}$.

98. In a $30^\circ-60^\circ-90^\circ$ triangle, what is the short leg if the hypotenuse is $h$?

    $\frac{h}{2}$. Short leg (opposite 30°) is half the hypotenuse in 30-60-90 triangles.

99. What is the converse of the Pythagorean Theorem for sides $a,b,c$ with $c$ largest?

    If $a^2+b^2=c^2$, then the triangle is right. If the sum of squares of two sides equals the square of the third, it's a right triangle.

100. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$, what is the long leg?

     $x\sqrt{3}$. Long leg equals short leg times $\sqrt{3}$ in a 30-60-90 triangle.

101. What is the long leg if a $30^\circ$-$60^\circ$-$90^\circ$ triangle has short leg $4$?

     $4\sqrt{3}$. Long leg = short leg × $\sqrt{3}$ in $30°$-$60°$-$90°$ triangle.

102. In a $45^\circ-45^\circ-90^\circ$ triangle, what is each leg if the hypotenuse is $h$?

     $\frac{h}{\sqrt{2}}$. Divide hypotenuse by $\sqrt{2}$ to find each equal leg.

103. State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $x$.

     legs $x,x$; hypotenuse $x\sqrt{2}$. Isosceles right triangle has hypotenuse $\sqrt{2}$ times the leg.

104. Identify the hypotenuse in a right triangle in terms of side length and angle position.

     The side opposite the $90^\circ$ angle; it is the longest side. The hypotenuse is always opposite the right angle.

105. Find the area of a right triangle with legs $a=12$ and $b=5$.

     $30$. Apply $A=\frac{1}{2}ab=\frac{1}{2}(12)(5)=\frac{60}{2}=30$.

106. State the side-length ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle.

     $1:1:\sqrt{2}$. Isosceles right triangles have equal legs and hypotenuse $\sqrt{2}$ times longer.

107. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$, what is the hypotenuse?

     $2x$. Hypotenuse is twice the short leg in a 30-60-90 triangle.

108. Find $c$ if a right triangle has legs $9$ and $12$.

     $15$. Apply Pythagorean theorem: $9^2+12^2=81+144=225=15^2$.

109. What is the hypotenuse in a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg length $x$?

     $x\sqrt{2}$. In isosceles right triangles, hypotenuse equals leg times $\sqrt{2}$.

110. What are the side lengths in the Pythagorean triple with legs $3$ and $4$?

     $3,4,5$. The hypotenuse is $\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$.

111. Identify which side is the hypotenuse in a right triangle relative to the $90^\circ$ angle.

     The side opposite the $90^\circ$ angle. The longest side in a right triangle, connecting the two legs.

112. Find the missing leg $b$ if a right triangle has hypotenuse $c=13$ and leg $a=5$.

     $b=12$. Use $a^2+b^2=c^2$: $5^2+b^2=13^2$, so $b^2=169-25=144$, thus $b=12$.

113. State the side-length ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle (short:long:hyp).

     $1:\sqrt{3}:2$. The sides are in proportion where hypotenuse is twice the shortest side.

114. Find the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $7$.

     $7\sqrt{2}$. Multiply leg by $\sqrt{2}$ in a 45-45-90 triangle.

115. Find the distance between $(1,2)$ and $(4,6)$ using a right-triangle distance calculation.

     $5$. $d=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=\sqrt{25}=5$.

116. State the area formula for a right triangle with legs $a$ and $b$ as base and height.

     $A=\frac{1}{2}ab$. The legs form the base and height, so area is half their product.

117. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the short leg if the hypotenuse is $16$?

     $8$. Short leg = hypotenuse ÷ 2 in 30-60-90 triangles.

118. Find the long leg of a $30^\circ-60^\circ-90^\circ$ triangle if the short leg is $4$.

     $4\sqrt{3}$. Long leg = short leg × $\sqrt{3}$ in 30-60-90 triangle: $4\times\sqrt{3}$.

119. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is a leg if the hypotenuse is $10\sqrt{2}$?

     $10$. Divide $10\sqrt{2}$ by $\sqrt{2}$ to get leg length.

120. Identify the missing leg: right triangle with hypotenuse $13$ and leg $5$; what is the other leg?

     $12$. Use $a^2 + b^2 = c^2$: $5^2 + b^2 = 13^2$, so $b = \sqrt{144} = 12$.

121. What is the relationship between slopes $m_1$ and $m_2$ for perpendicular lines?

     $m_1m_2=-1$. Perpendicular lines have slopes whose product is $-1$.

122. Find the distance between $(0,0)$ and $(3,4)$.

     $5$. Distance $=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$.

123. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is the hypotenuse if each leg is $x$?

     $x\sqrt{2}$. Apply the $1:1:\sqrt{2}$ ratio with legs of length $x$.

124. Find the missing leg $b$ if $c=13$ and $a=5$ in a right triangle.

     $12$. Use $b=\sqrt{c^2-a^2}=\sqrt{169-25}=\sqrt{144}=12$.

125. Find the missing leg length if the hypotenuse is $13$ and one leg is $5$.

     $12$. Use $a^2+b^2=c^2$: $5^2+b^2=13^2$, so $b^2=169-25=144$, thus $b=12$.

126. State the relationship between the two acute angles in any right triangle.

     They are complementary: sum is $90^\circ$. The two non-right angles must add to $90°$ in any right triangle.

127. Identify which side is the hypotenuse in a right triangle.

     The side opposite the $90^\circ$ angle. The longest side in a right triangle, always opposite the right angle.

128. What are the side lengths in the Pythagorean triple with legs $8$ and $15$?

     $8,15,17$. The hypotenuse is $\sqrt{8^2+15^2}=\sqrt{64+225}=\sqrt{289}=17$.

129. What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ from the Pythagorean Theorem?

     $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Applies Pythagorean theorem to coordinate differences.

130. What is the distance between $(x_1,y_1)$ and $(x_2,y_2)$ using a right-triangle interpretation?

     $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Forms a right triangle with legs as coordinate differences.

131. Identify whether $8,15,17$ can be side lengths of a right triangle using the Pythagorean Theorem.

     Yes, because $8^2+15^2=17^2$. Check: $64 + 225 = 289$, which equals $17^2$.

132. Find the distance between $(1,2)$ and $(4,6)$.

     $5$. $d=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=\sqrt{25}=5$.

133. Find the distance between $(1,2)$ and $(4,6)$ using a right triangle.

     $5$. Use $d=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=\sqrt{25}=5$.

134. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, find each leg if the hypotenuse is $10\sqrt{2}$.

     $10$. Divide hypotenuse by $\sqrt{2}$ to get each equal leg.

135. State the area formula for a right triangle with legs $a$ and $b$.

     $A=\frac{1}{2}ab$. Half the product of the two legs gives the area.

136. Identify the missing side: right triangle with legs $6$ and $8$; what is the hypotenuse $c$?

     $10$. Apply Pythagorean theorem: $c = \sqrt{6^2 + 8^2} = \sqrt{100} = 10$.

137. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, find the long leg if the short leg is $4$.

     $4\sqrt{3}$. Long leg equals short leg times $\sqrt{3}$ in 30-60-90 triangles.

138. Find the missing side $x$ if a right triangle has legs $x$ and $12$ and hypotenuse $15$.

     $9$. Use $x=\sqrt{15^2-12^2}=\sqrt{225-144}=\sqrt{81}=9$.

139. Find the slope of the line through $(2,3)$ and $(6,1)$ using $m=\frac{y_2-y_1}{x_2-x_1}$.

     $-\frac{1}{2}$. $m=\frac{1-3}{6-2}=\frac{-2}{4}=-\frac{1}{2}$.

140. Find $x$ if a right triangle has legs $x$ and $24$ and hypotenuse $25$.

     $7$. Using $x^2+24^2=25^2$, so $x^2=625-576=49$, thus $x=7$.

141. Identify which side is the hypotenuse in a right triangle relative to the $90^\circ$ angle.

     The side opposite the $90^\circ$ angle. The longest side in a right triangle, opposite the right angle.

142. Find the hypotenuse $c$ if the legs are $6$ and $8$.

     $10$. Apply Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.

143. State the inequality test for an obtuse triangle with largest side $c$ and other sides $a,b$.

     Obtuse if $a^2+b^2<c^2$. When the sum of squares is less than $c^2$, the largest angle is obtuse.

144. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the side opposite $60^\circ$ if the short leg is $x$?

     $x\sqrt{3}$. Long leg equals short leg times $\sqrt{3}$ in 30-60-90 triangles.

145. Identify the hypotenuse in a right triangle in terms of the right angle.

     The side opposite the $90^\circ$ angle. The hypotenuse is always the longest side in a right triangle.

146. Find the long leg if a $30^\circ$-$60^\circ$-$90^\circ$ triangle has short leg $4$.

     $4\sqrt{3}$. Long leg = short leg × $\sqrt{3}$ in a 30-60-90 triangle.

147. Find the slope of a line perpendicular to a line with slope $\frac{3}{4}$.

     $-\frac{4}{3}$. Perpendicular slope is negative reciprocal: $-1÷\frac{3}{4}$.

148. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is the hypotenuse if a leg is $7$?

     $7\sqrt{2}$. Multiply leg by $\sqrt{2}$ for 45-45-90 hypotenuse.

149. State the side ratio for a $30^\circ-60^\circ-90^\circ$ right triangle in simplest radical form.

     $1:\sqrt{3}:2$. Short leg : long leg : hypotenuse ratio for 30-60-90 triangles.

150. Find the missing leg $a$ if $c=13$ and $b=5$ in a right triangle.

     $12$. Rearrange Pythagorean theorem: $a=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.

151. State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle in simplest form.

     $1:1:\sqrt{2}$. Isosceles right triangle has equal legs and hypotenuse $\sqrt{2}$ times longer.

152. Find the hypotenuse: right triangle with legs $9$ and $12$; what is $c$?

     $15$. Calculate $c = \sqrt{9^2 + 12^2} = \sqrt{225} = 15$.

153. What is the $45^\circ$-$45^\circ$-$90^\circ$ side ratio (leg, leg, hypotenuse)?

     $1:1:\sqrt{2}$. In an isosceles right triangle, legs are equal and hypotenuse is $\sqrt{2}$ times a leg.

154. What is the formula for the area of a right triangle with legs $a$ and $b$?

     $A=\frac{1}{2}ab$. Half the product of the two perpendicular sides (legs).

155. Find the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $9$.

     $9\sqrt{2}$. In a $45°$-$45°$-$90°$ triangle, hypotenuse = leg × $\sqrt{2}$.

156. What is the converse of the Pythagorean Theorem for side lengths $a,b,c$ with $c$ largest?

     If $a^2+b^2=c^2$, then the triangle is right. If the equation holds, the angle opposite side $c$ is $90°$.

157. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is a leg if the hypotenuse is $h$?

     $\frac{h}{\sqrt{2}}$. Divide hypotenuse by $\sqrt{2}$ to find each equal leg.

158. Find the area of a right triangle with legs $10$ and $12$.

     $60$. Use $A=\frac{1}{2}ab=\frac{1}{2}(10)(12)=60$.

159. What is the sine ratio in a right triangle for acute angle $\theta$?

     $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Ratio of the side opposite angle $\theta$ to the hypotenuse.

160. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the hypotenuse if the short leg is $x$?

     $2x$. Hypotenuse is always twice the short leg in 30-60-90 triangles.

161. Find the distance between $(1,2)$ and $(4,6)$ using the distance formula.

     $5$. $\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=\sqrt{25}=5$.

162. Find $c$ if a right triangle has legs $a=6$ and $b=8$.

     $c=10$. Apply $a^2+b^2=c^2$: $6^2+8^2=36+64=100$, so $c=\sqrt{100}=10$.

163. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the long leg if the short leg is $x$?

     $x\sqrt{3}$. Long leg = short leg × $\sqrt{3}$ in 30-60-90 triangles.

164. Find the long leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x=4$.

     $4\sqrt{3}$. In 30-60-90 triangle, long leg = short leg × $\sqrt{3}$.

165. What is the longer leg if a $30^\circ$-$60^\circ$-$90^\circ$ triangle has shorter leg $x$?

     $x\sqrt{3}$. The longer leg (opposite $60°$) is $\sqrt{3}$ times the shorter leg.

166. What is the relationship between the acute angles $A$ and $B$ in a right triangle?

     $A+B=90^\circ$. The two non-right angles are complementary.

167. Which inequality classifies a triangle as obtuse using sides $a\le b<c$?

     Obtuse if $a^2+b^2<c^2$. When leg squares are less than hypotenuse square, one angle exceeds $90°$.

168. Find the long leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $5$.

     $5\sqrt{3}$. Multiply short leg by $\sqrt{3}$ to get long leg.

169. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the side opposite $30^\circ$ if the hypotenuse is $2x$?

     $x$. Short leg is always half the hypotenuse in 30-60-90 triangles.

170. Find the area of a right triangle with legs $9$ and $12$.

     $54$. Area $=\frac{1}{2}\times 9\times 12=\frac{108}{2}=54$.

171. State the side ratios for a $45^\circ$-$45^\circ$-$90^\circ$ triangle (legs and hypotenuse).

     $x,\ x,\ x\sqrt{2}$. Isosceles right triangle has equal legs and hypotenuse $\sqrt{2}$ times longer.

172. Find the hypotenuse if a $45^\circ$-$45^\circ$-$90^\circ$ triangle has leg $7$.

     $7\sqrt{2}$. In $45°$-$45°$-$90°$ triangle, hypotenuse = leg × $\sqrt{2}$.

173. What is the hypotenuse if a $45^\circ$-$45^\circ$-$90^\circ$ triangle has leg $6$?

     $6\sqrt{2}$. In $45°$-$45°$-$90°$ triangle, hypotenuse = leg × $\sqrt{2}$.

174. Identify whether sides $7,24,25$ form a right triangle using $a^2+b^2=c^2$.

     Yes, because $7^2+24^2=25^2$. Check: $49+576=625$, which equals $25^2$.

175. Identify whether side lengths $9,12,15$ form a right triangle.

     Yes, because $9^2+12^2=15^2$. Check if $9^2+12^2=15^2$: $81+144=225$, which is true.

176. Identify the hypotenuse in a right triangle in terms of its relationship to the right angle.

     The side opposite the $90^\circ$ angle. The hypotenuse is always opposite the right angle in any right triangle.

177. State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg length $x$.

     Sides $x,x,x\sqrt{2}$. In isosceles right triangles, hypotenuse = leg × $\sqrt{2}$.

178. What is the converse of the Pythagorean Theorem for side lengths $a,b,c$ with $c$ largest?

     If $a^2+b^2=c^2$, the triangle is right. If the equation holds, then the triangle must have a right angle.

179. State the special right triangle ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle.

     Side ratio $1:1:\sqrt{2}$. In an isosceles right triangle, legs are equal and hypotenuse is $\sqrt{2}$ times a leg.

180. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is the hypotenuse in terms of a leg $x$?

     $x\sqrt{2}$. In isosceles right triangles, multiply leg by $\sqrt{2}$ to get hypotenuse.

181. In a $45^\circ-45^\circ-90^\circ$ triangle, what is the hypotenuse if a leg is $x$?

     $x\sqrt{2}$. In a 45-45-90 triangle, hypotenuse = leg × $\sqrt{2}$.

182. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the long leg if the short leg is $4$?

     $4\sqrt{3}$. Multiply short leg by $\sqrt{3}$ for long leg in 30-60-90.

183. Find $c$ if a right triangle has legs $a=6$ and $b=8$.

     $c=10$. Apply Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}$.

184. What is the area formula for a right triangle with legs $a$ and $b$?

     $A=\frac{1}{2}ab$. The legs serve as base and height in a right triangle.

185. What is the length of the hypotenuse if the legs are $6$ and $8$?

     $10$. Apply Pythagorean theorem: $\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.

186. Find the slope of a line perpendicular to a line with slope $\frac{2}{3}$.

     $-\frac{3}{2}$. Take negative reciprocal: $-\frac{1}{\frac{2}{3}}=-\frac{3}{2}$.

187. What is the cosine ratio in a right triangle for acute angle $\theta$?

     $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Ratio of the side adjacent to angle $\theta$ to the hypotenuse.

188. State the Pythagorean Theorem relating legs $a,b$ and hypotenuse $c$ in a right triangle.

     $a^2+b^2=c^2$. Relates the squares of the legs to the square of the hypotenuse.

189. What is the converse of the Pythagorean Theorem using side lengths $a\le b < c$?

     If $a^2+b^2=c^2$, then the triangle is right. If the sum of squares of two sides equals the square of the third, it's a right triangle.

190. What is the definition of the hypotenuse in a right triangle?

     The side opposite the $90^\circ$ angle. The longest side in any right triangle.

191. What is the hypotenuse in a $45^\circ$-$45^\circ$-$90^\circ$ triangle with legs $x$?

     $x\sqrt{2}$. In a 45-45-90 triangle, multiply leg by $\sqrt{2}$ for hypotenuse.

192. Classify a triangle with sides $6,8,10$ using the Pythagorean Theorem.

     Right triangle. $6^2 + 8^2 = 36 + 64 = 100 = 10^2$, so it's right.

193. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the long leg if the short leg is $5$?

     $5\sqrt{3}$. Long leg = short leg × $\sqrt{3}$ in a $30°$-$60°$-$90°$ triangle.

194. Find the missing leg $a$ if $c=13$ and the other leg is $b=5$.

     $a=12$. Rearrange Pythagorean theorem: $a=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}$.

195. What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ from the Pythagorean Theorem?

     $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Applies Pythagorean theorem to coordinate differences.

196. What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ derived from a right triangle?

     $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses the Pythagorean theorem on the coordinate differences.

197. Decide if sides $7,24,25$ form a right triangle using $a^2+b^2=c^2$.

     Yes, because $7^2+24^2=25^2$. Check: $7^2+24^2=49+576=625=25^2$, confirming it's a right triangle.

198. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is the hypotenuse if a leg is $x$?

     $x\sqrt{2}$. In isosceles right triangles, hypotenuse = leg × $\sqrt{2}$.

199. Find the long leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $7$.

     $7\sqrt{3}$. Multiply short leg by $\sqrt{3}$ to get long leg in $30°$-$60°$-$90°$ triangle.

200. In a right triangle, if the hypotenuse is $13$ and one leg is $5$, what is the other leg?

     $12$. Use $5^2+b^2=13^2$: $25+b^2=169$, so $b^2=144$ and $b=12$.

201. State the area formula for a right triangle with legs $a$ and $b$.

     $A=\frac{1}{2}ab$. The legs form a rectangle with area $ab$; the triangle is half of that.

202. State the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ derived from right triangles.

     $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses Pythagorean theorem on the horizontal and vertical distances.

203. State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle (short:long:hyp).

     $1:\sqrt{3}:2$. Special triangle with sides in ratio based on $30°$, $60°$, and $90°$ angles.

204. What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ derived from right triangles?

     $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses the Pythagorean theorem on the coordinate plane.

205. Identify whether sides $7,24,25$ form a right triangle using $a^2+b^2=c^2$.

     Yes, it is a right triangle. Check: $7^2+24^2=49+576=625=25^2$, so it satisfies Pythagorean theorem.

206. Find the missing leg $a$ if $c=13$ and the other leg is $5$.

     $12$. Using Pythagorean theorem: $a=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.

207. Identify the hypotenuse in a right triangle in terms of the right angle.

     The side opposite the $90^\circ$ angle. The hypotenuse is always the longest side in a right triangle.

208. Find the distance between points $(1,2)$ and $(7,10)$.

     $10$. Apply distance formula: $d=\sqrt{(7-1)^2+(10-2)^2}=\sqrt{36+64}=10$.

209. What is the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be perpendicular?

     $m_1m_2=-1$. Perpendicular lines have slopes that are negative reciprocals.

210. Classify a triangle with sides $4,5,6$ as acute, right, or obtuse using $a^2+b^2$ vs. $c^2$.

     Acute triangle. $4^2 + 5^2 = 41 > 36 = 6^2$, so all angles are acute.

211. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what side is opposite the $30^\circ$ angle?

     The short leg $x$. Smallest angle faces smallest side in any triangle.

212. Find the short leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle with hypotenuse $18$.

     $9$. In a $30°$-$60°$-$90°$ triangle, short leg = hypotenuse ÷ 2.

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