Triangles - ACT Math

1. What is the definition of a right triangle in terms of angles?

   One angle is $90^\circ$. A right angle defines a right triangle.

2. What is the condition for a triangle to be acute using side lengths with largest side $c$?

   $a^2+b^2>c^2$. When legs squared exceed hypotenuse squared, all angles are acute.

3. What is the side opposite the right angle in a right triangle called?

   Hypotenuse. The longest side in a right triangle, opposite the right angle.

4. What is the midpoint of the segment with endpoints $(2,4)$ and $(8,10)$?

   $(5,7)$. $\left(\frac{2+8}{2}, \frac{4+10}{2}\right) = (5,7)$

5. What is the slope of the line through points $ (1,2) $ and $ (5,10) $?

   $2$. $\frac{10-2}{5-1} = \frac{8}{4} = 2$

6. What is the sum of interior angles in any triangle?

   180 degrees. This is a fundamental property that applies to all triangles.

7. What is the relationship between an exterior angle and its remote interior angles?

   Exterior angle = sum of remote interior angles. Exterior angle equals sum of the two non-adjacent interior angles.

8. What is the altitude in an equilateral triangle with side $s$?

   $\frac{\sqrt{3}}{2} \times s$. Height of equilateral triangle using special right triangle ratios.

9. What is the formula for the area of an equilateral triangle with side $s$?

   $\frac{\sqrt{3}}{4} \times s^2$. Derived from the general triangle area formula for equilateral triangles.

10. What is the definition of an isosceles triangle in terms of side lengths?

    At least two sides are congruent. Two or more equal sides define an isosceles triangle.

11. What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

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

12. Find the length of the longer leg in a 30-60-90 triangle with short leg 5.

    $5 \sqrt{3}$. In 30-60-90 triangle, long leg is short leg times $\sqrt{3}$.

13. Given a triangle with sides 7, 24, 25, classify it.

    Right triangle. Check: $7^2 + 24^2 = 49 + 576 = 625 = 25^2$, so right triangle.

14. What is each interior angle of an equilateral triangle?

    $60^\circ$. $180^\circ \div 3 = 60^\circ$ for each equal angle.

15. What is the definition of an equilateral triangle in terms of side lengths?

    All three sides are congruent. Equal sides create an equilateral triangle.

16. What is the area of a right triangle in terms of its legs $a$ and $b$?

    $A=\frac{1}{2}ab$. Right triangles use legs as base and height.

17. What is the orthocenter of a triangle?

    The point where the altitudes meet. Intersection point of all three altitude lines.

18. Identify the type of triangle with angles 90°, 45°, 45°.

    Isosceles right triangle. Right triangle with two equal legs and equal base angles.

19. What type of triangle has no sides of equal length?

    Scalene triangle. All three sides have different lengths.

20. What is the perimeter of a triangle with side lengths $7$, $8$, and $9$?

    $24$. $7 + 8 + 9 = 24$

21. What is the Heron's formula for the area of a triangle?

    $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$. Calculates area using only the three side lengths.

22. What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$ of a segment?

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the x-coordinates and y-coordinates separately.

23. What is the missing leg of a right triangle with hypotenuse $13$ and one leg $5$?

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

24. What is the special right triangle ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle?

    $1:1:\sqrt{2}$. Isosceles right triangle has hypotenuse $\sqrt{2}$ times leg.

25. Identify the missing angle: a triangle has angles $50^\circ$ and $60^\circ$; what is the third angle?

    $70^\circ$. $180^\circ - 50^\circ - 60^\circ = 70^\circ$

26. Determine the longest side in a triangle with sides 7, 10, 5.

    10. The longest side is opposite the largest angle.

27. What is the definition of a right triangle?

    A triangle with one 90-degree angle. Contains exactly one angle measuring $90°$.

28. What is an altitude in a triangle?

    A perpendicular segment from a vertex to the opposite side. Creates the shortest distance from vertex to opposite side.

29. Find the hypotenuse in a 30-60-90 triangle with short leg 5.

    10. In 30-60-90 triangle, hypotenuse is twice the short leg.

30. What is the measure of each angle in an equilateral triangle?

    60 degrees. Since all sides are equal, all angles are equal: $180° ÷ 3 = 60°$.

31. What is the altitude in a triangle?

    A perpendicular segment from a vertex to the opposite side. Height drops perpendicularly from vertex to opposite side.

32. What is the hypotenuse of a right triangle with legs $6$ and $8$?

    $10$. $\sqrt{6^2 + 8^2} = \sqrt{100} = 10$

33. What is the name of a triangle with angles 60°, 60°, 60°?

    Equilateral triangle. All angles are equal, so each measures $60°$.

34. Identify whether side lengths $2$, $3$, and $6$ can form a triangle using the triangle inequality.

    No, it cannot form a triangle. $2 + 3 = 5 < 6$, violating triangle inequality.

35. Identify the triangle inequality theorem.

    The sum of any two sides must be greater than the third side. Ensures a valid triangle can be formed with three given lengths.

36. Find the length of the hypotenuse: legs are 3 and 4.

    5. Using Pythagorean theorem: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

37. What is the definition of an exterior angle in a triangle?

    An angle formed by one side and the extension of another. Formed by extending one side beyond a vertex.

38. Calculate the area of a triangle with base 10 and height 5.

    25 square units. Using $\frac{1}{2} \times 10 \times 5 = 25$.

39. What is the slope formula for points $ ($x_1$, $y_1$) $ and $ ($x_2$, $y_2$) $?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points.

40. What is the centroid of a triangle?

    The point where the medians meet. Balance point where triangle would rest on a pin.

41. State the Cosine Rule for triangle sides $a$, $b$, $c$ and angle $C$.

    $c^2 = a^2 + b^2 - 2ab \cos C$. General formula connecting all sides and one angle.

42. What is the definition of a congruent triangle?

    Triangles with equal corresponding angles and sides. Triangles that are identical in shape and size.

43. Define an isosceles triangle.

    A triangle with two equal sides. Two sides have equal length, creating two equal angles.

44. What do the exterior angles of a triangle sum to?

    360°. Each exterior angle plus its interior angle equals $180°$.

45. Find the third angle in a triangle with angles 35° and 65°.

    80°. Sum of angles is $180°$: $180° - 35° - 65° = 80°$.

46. Identify whether sides $3$, $4$, and $5$ form a right triangle using $a^2+b^2=c^2$.

    Yes, it is a right triangle. $3^2 + 4^2 = 9 + 16 = 25 = 5^2$

47. What is the distance formula for the hypotenuse when legs are $a$ and $b$ in a right triangle?

    $c=\sqrt{a^2+b^2}$. Take the square root of the sum of squared legs.

48. What is the semi-perimeter of a triangle with sides $a$, $b$, $c$?

    $\frac{a+b+c}{2}$. Half the total perimeter, used in area formulas.

49. State the formula for the area of an equilateral triangle with side $s$.

    $\frac{\sqrt{3}}{4} s^2$. Derived from base-height formula using equilateral properties.

50. Calculate the perimeter of a triangle with sides 3, 4, 5.

    12. Sum of all three sides: $3 + 4 + 5 = 12$.

51. Define a scalene triangle.

    A triangle with all sides of different lengths. No two sides have the same length.

52. What type of triangle has all sides of equal length?

    Equilateral triangle. All three sides are congruent in this regular triangle.

53. Which formula represents the Law of Sines?

    $\frac{a}{\text{sin } A} = \frac{b}{\text{sin } B} = \frac{c}{\text{sin } C}$. Relates sides to opposite angles in any triangle.

54. Find the hypotenuse in a 45-45-90 triangle with legs of length 7.

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

55. What is the incenter of a triangle?

    The point where the angle bisectors meet. Center of the circle inscribed within the triangle.

56. What is the sum of the exterior angles of any triangle?

    360 degrees. Constant sum regardless of triangle shape or size.

57. What is the definition of an acute triangle in terms of angles?

    All angles are less than $90^\circ$. All angles are sharp in an acute triangle.

58. What is the Triangle Inequality Theorem stated in terms of side lengths $a$, $b$, and $c$?

    $a+b>c$, $a+c>b$, and $b+c>a$. Each side must be less than the sum of the other two sides.

59. Identify the angle bisector theorem.

    Divides the opposite side into segments proportional to the other two sides. Angle bisector creates proportional segments on the opposite side.

60. What is the area of a triangle with base $10$ and height $6$?

    $30$. $\frac{1}{2} \times 10 \times 6 = 30$

61. What is the definition of an obtuse triangle in terms of angles?

    One angle is greater than $90^\circ$. One wide angle defines an obtuse triangle.

62. What is the Pythagorean theorem?

    $a^2 + b^2 = c^2$. Relates the sides of a right triangle: legs squared sum equals hypotenuse squared.

63. What is the hypotenuse if the legs of a $45^\circ$-$45^\circ$-$90^\circ$ triangle are $x$ and $x$?

    $x\sqrt{2}$. Multiply leg by $\sqrt{2}$ for the hypotenuse.

64. What is the sum of interior angles in a triangle?

    180°. Sum of all angles in any triangle is always constant.

65. Identify the hypotenuse in a right triangle with sides 3, 4, 5.

    5. The hypotenuse is the longest side in a right triangle.

66. What is the circumcenter of a triangle?

    The point where the perpendicular bisectors of the sides meet. Center of the circle passing through all three vertices.

67. Identify the type of triangle with all sides equal.

    Equilateral triangle. All three sides are identical in length.

68. State the Pythagorean Theorem.

    $a^2 + b^2 = c^2$. Relates the squares of sides in a right triangle.

69. What is the special right triangle ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle?

    $1:\sqrt{3}:2$. Short leg, long leg, hypotenuse in the $30^\circ$-$60^\circ$-$90^\circ$ triangle.

70. State the Sine Rule for triangle sides $a$, $b$, $c$ and angles $A$, $B$, $C$.

    $\frac{a}{\text{sin} A} = \frac{b}{\text{sin} B} = \frac{c}{\text{sin} C}$. Relates side lengths to their opposite angles via ratios.

71. Find the third side of a triangle with sides 8 and 15, and an angle of 90° between them.

    17. Using Pythagorean theorem: $\sqrt{8^2 + 15^2} = \sqrt{289} = 17$.

72. What is the perimeter of a triangle with side lengths $a$, $b$, and $c$?

    $P=a+b+c$. Add all three side lengths together.

73. State the formula for the area of a triangle.

    $\frac{1}{2} \times \text{base} \times \text{height}$. Standard formula where height is perpendicular distance to the base.

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

    $x\sqrt{3}$. Multiply shorter leg by $\sqrt{3}$ for longer leg.

75. What is the distance between points $(0,0)$ and $(3,4)$?

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

76. What is the condition for a triangle to be right using side lengths with largest side $c$?

    $a^2+b^2=c^2$. When this equation holds, the triangle has a $90^\circ$ angle.

77. What is a median in a triangle?

    A segment from a vertex to the midpoint of the opposite side. Connects vertex to midpoint of opposite side.

78. Find the third angle in a triangle with angles 45° and 75°.

    60 degrees. Sum of angles is $180°$: $180° - 45° - 75° = 60°$.

79. Identify whether sides $5$, $6$, and $7$ form an acute, right, or obtuse triangle.

    Acute. $5^2 + 6^2 = 61 > 49 = 7^2$, so acute.

80. What is true about base angles in an isosceles triangle?

    The base angles are congruent. Angles opposite equal sides are equal.

81. Identify the type of triangle with sides 5, 5, and 8.

    Isosceles triangle. Two sides are equal ($5$ each), third side different.

82. What is the sum of the interior angles of any triangle in degrees?

    $180^\circ$. The angles in any triangle always add up to this value.

83. Which triangle inequality must hold for any triangle?

    The sum of any two sides must be greater than the third side. Ensures triangle can actually be formed from three sides.

84. Which triangle has one angle greater than 90°?

    Obtuse triangle. One angle exceeds $90°$, making it wider than a right angle.

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

    $2x$. Hypotenuse is twice the shorter leg.

86. What is the side ratio in a 45-45-90 triangle?

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

87. What is the formula for the semi-perimeter of a triangle?

    $s = \frac{a + b + c}{2}$. Half of the perimeter, used in Heron's formula.

88. What is the definition of a scalene triangle in terms of side lengths?

    No sides are congruent. All sides have different lengths in a scalene triangle.

89. Which formula represents the Law of Cosines?

    $c^2 = a^2 + b^2 - 2ab \times \cos C$. Generalizes Pythagorean theorem for any triangle using cosine.

90. What is the formula for the area of a triangle?

    $\frac{1}{2} \times \text{base} \times \text{height}$. Half the product of base and perpendicular height.

91. What type of triangle has two sides of equal length?

    Isosceles triangle. Two sides are equal, making two base angles equal as well.

92. Find the missing angle if two angles are 45° and 60°.

    75°. Sum of angles is $180°$: $180° - 45° - 60° = 75°$.

93. What is the formula for the perimeter of a triangle?

    Sum of all side lengths. Add all three side lengths: $a + b + c$.

94. What is the Pythagorean Theorem for a right triangle with legs $a$, $b$ and hypotenuse $c$?

    $a^2+b^2=c^2$. Sum of squares of legs equals square of hypotenuse.

95. What is the condition for a triangle to be obtuse using side lengths with largest side $c$?

    $a^2+b^2<c^2$. When legs squared are less than hypotenuse squared, one angle is obtuse.

96. What is the area formula for a triangle with base $b$ and height $h$?

    $A=\frac{1}{2}bh$. Base times height divided by 2 gives triangle area.

97. What is the hypotenuse in a right triangle?

    The side opposite the right angle. The longest side in a right triangle, opposite the $90°$ angle.

98. What is the shorter leg if the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle is $x$?

    $\frac{x}{\sqrt{2}}$. Divide hypotenuse by $\sqrt{2}$ for each leg.

99. What is the angle opposite the largest side in a triangle?

    Largest angle. In any triangle, the largest angle faces the longest side.

100. Find the area of a right triangle with legs 6 and 8.

     $24$ square units. Using $\frac{1}{2} \times 6 \times 8 = 24$ for right triangle legs.

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