Explain Pythagorean Theorem Proof - 8th Grade Math

1. Which side must be used as $c$ when applying $a^2+b^2=c^2$ to a triangle?

   The longest side (the hypotenuse if the triangle is right). The theorem requires $c$ to be the hypotenuse for validity.

2. Which equation tests whether a triangle with sides $a,b,c$ (with $c$ largest) is right?

   Check whether $a^2+b^2=c^2$. The converse tests if a triangle is right by checking this equality.

3. Identify the error: A student uses $a^2+c^2=b^2$ for a right triangle with hypotenuse $c$.

   Correct equation: $a^2+b^2=c^2$ (hypotenuse must be alone). The theorem requires $c$ to be the hypotenuse, not a leg.

4. What is $a$ if a right triangle has hypotenuse $c=13$ and leg $b=5$?

   $a=12$. Use $a^2=13^2-5^2=169-25=144$, so $a=sqrt{144}=12$.

5. After expanding $(a+b)^2=a^2+2ab+b^2$ and setting it equal to $2ab+c^2$, what conclusion follows?

   $a^2+b^2=c^2$. Cancel $2ab$ from both sides to get the Pythagorean theorem.

6. What does the Pythagorean Theorem say about the side labeled $c$ in $a^2+b^2=c^2$?

   $c$ is the hypotenuse (side opposite the right angle). In a right triangle, $c$ must be the longest side.

7. What is the correct conclusion if a triangle has side lengths $6,8,10$?

   It is a right triangle because $6^2+8^2=10^2$. Check: $36+64=100$, which equals $10^2$.

8. Find $c$ if a right triangle has legs $a=3$ and $b=4$.

   $c=5$. Apply theorem: $3^2+4^2=9+16=25$, so $c=sqrt{25}=5$.

9. What simplified equation results from $(a+b)^2=2ab+c^2$ after expanding $(a+b)^2$?

   $a^2+b^2=c^2$. Expand $(a+b)^2=a^2+2ab+b^2$, then subtract $2ab$ from both sides.

10. What is the converse of the Pythagorean Theorem using side lengths $a,b,c$ with $c$ longest?

    If $a^2+b^2=c^2$, then the triangle is right. Reverses the theorem: if the equation holds, then the angle is 90°.

11. What does the area model proof compare to show $a^2+b^2=c^2$?

    Two expressions for the same square's area. The proof shows $(a+b)^2 = 4$ triangles $+ c^2$ square.

12. Identify the missing term: Expanding $(a+b)^2$ gives $a^2+\_+b^2$.

    $2ab$. The binomial expansion includes the cross term $2ab$.

13. What is $c$ if a right triangle has legs $a=9$ and $b=12$?

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

14. What does the symbol $c$ represent in $a^2+b^2=c^2$ for a right triangle?

    $c$ is the hypotenuse (longest side). In right triangles, the hypotenuse is always opposite the right angle.

15. What is $c$ if $a^2+b^2=625$ and $c>0$ for a right triangle?

    $c=25$. Since $c^2=625$, take the positive square root.

16. Use the converse: Are side lengths $6,8,10$ a right triangle (yes or no)?

    Yes, because $6^2+8^2=10^2$. Check: $36+64=100$ is true, confirming a right triangle.

17. Which substitution is correct in a proof: $c$ equals the side opposite the $90^\circ$ angle or adjacent?

    $c$ equals the side opposite the $90^\circ$ angle. The hypotenuse is always opposite the right angle.

18. Find the missing leg $a$ if $c=10$ and $b=6$ in a right triangle.

    $a=8$. Rearrange: $a^2=c^2-b^2=100-36=64$, so $a=8$.

19. What is the area formula for a square with side length $s$ used in Pythagorean proofs?

    $A=s^2$. Square area equals side length squared.

20. Identify whether sides $6,8,10$ form a right triangle using the converse.

    Yes, because $6^2+8^2=10^2$. Check: $36+64=100$ ✓, confirming a right angle.

21. What equation results from equating the two area expressions in the square proof?

    $(a+b)^2=2ab+c^2$. Large square = 4 triangles + inner square.

22. In the square proof, what is the area of the central square if its side length is $c$?

    $c^2$. The hypotenuse forms the side of the inner square.

23. What is the area expression for the large square of side $a+b$ in the rearrangement proof?

    $(a+b)^2$. Expands to $a^2+2ab+b^2$ using the binomial formula.

24. What is the area of a square with side length $s$ (used in some Pythagorean proofs)?

    $s^2$. Area equals side length squared for any square.

25. What is the correct conclusion if a triangle has side lengths $5,6,7$?

    It is not a right triangle because $5^2+6^2\neq7^2$. Check: $25+36=61$, but $7^2=49$, so $61≠49$.

26. What conclusion is justified if $a^2+b^2<c^2$ for sides $a,b,c$ with $c$ largest?

    The triangle is obtuse. When the sum of leg squares is less than hypotenuse squared, angle exceeds 90°.

27. Identify whether sides $3,4,5$ form a right triangle using the converse.

    Yes, because $3^2+4^2=5^2$. Check: $9+16=25$ ✓, so the triangle has a right angle.

28. Identify the first step to use the converse: which side should be chosen as $c$?

    Choose the longest side as $c$. The converse requires comparing with the longest side squared.

29. Find the value of $a^2+b^2$ for legs $a=7$ and $b=24$ in a right triangle check.

    $625$. Calculate $7^2+24^2=49+576=625$.

30. In the square proof, what is the total area of the four congruent right triangles with legs $a$ and $b$?

    $4\big(\frac{1}{2}ab\big)=2ab$. Four triangles each with area $\frac{1}{2}ab$ sum to $2ab$.

31. Use the converse: Are side lengths $4,5,6$ a right triangle (yes or no)?

    No, because $4^2+5^2\ne6^2$. Check: $16+25=41 eq 36$, so not a right triangle.

32. What is the area of a right triangle with legs $a$ and $b$?

    $\frac{1}{2}ab$. Uses the formula: base times height divided by 2.

33. In the square proof, what is the area of the large square with side length $a+b$?

    $(a+b)^2$. The large square has side length equal to the sum of the two legs.

34. What is the key final statement of an area-based proof after cancellation and simplification?

    $a^2+b^2=c^2$. Area-based proofs ultimately derive this fundamental relationship.

35. Identify the triangle type if $a^2+b^2>c^2$ for sides $a,b,c$ with $c$ longest.

    Acute triangle. When leg squares exceed hypotenuse squared, all angles are less than 90°.

36. What is the area formula for a right triangle with legs $a$ and $b$ used in area proofs?

    $A=\frac{1}{2}ab$. Right triangle area is half the product of its legs.

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

    $b=12$. Rearrange: $b^2=c^2-a^2=169-25=144$, so $b=12$.

38. What is the area of a square built on a leg of length $a$ in a Pythagorean proof diagram?

    $a^2$. Each square's area equals the square of its side length.

39. In the common rearrangement proof, what is the side length of the large square built from legs $a$ and $b$?

    $a+b$. The outer square has sides equal to the sum of the two legs.

40. Identify the correct first step to use the converse when given three side lengths.

    Let $c$ be the longest side, then test whether $a^2+b^2=c^2$. The converse requires checking if the longest side satisfies the theorem.

41. Identify whether sides $5,6,7$ form a right triangle using the converse.

    No, because $5^2+6^2\ne7^2$. Check: $25+36=61≠49$, so no right angle exists.

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

    $a^2+b^2=c^2$. States that the sum of squares of the legs equals the square of the hypotenuse.

43. Which statement is correct for a right triangle: $a^2+b^2=c^2$ or $a+b=c$?

    $a^2+b^2=c^2$. The Pythagorean theorem uses squares, not simple addition.

44. Identify the error: A student used $7^2+24^2=25^2$ but labeled $7$ as the hypotenuse. What is the correction?

    The hypotenuse is $25$ (the longest side), not $7$. In the theorem, $c$ must be the longest side to be the hypotenuse.

45. Find $c$ for a right triangle with legs $a=9$ and $b=12$.

    $c=15$. Apply theorem: $c^2=81+144=225$, so $c=\sqrt{225}=15$.

46. What conclusion is justified if $a^2+b^2>c^2$ for sides $a,b,c$ with $c$ largest?

    The triangle is acute. When the sum of leg squares exceeds hypotenuse squared, all angles are under 90°.

47. What identity is used to expand $(a+b)^2$ in the area proof?

    $(a+b)^2=a^2+2ab+b^2$. Standard binomial expansion: square of sum equals sum of squares plus twice the product.

48. What does the Converse of the Pythagorean Theorem state about a triangle with sides $a,b,c$?

    If $a^2+b^2=c^2$, the triangle is right with hypotenuse $c$. A triangle is right-angled when the sum of squares of two sides equals the square of the third.

49. In a rearrangement proof, what is the area of a right triangle with legs $a$ and $b$?

    $\frac{1}{2}ab$. Uses the standard triangle area formula: base times height divided by 2.

50. Which statement is always true for a right triangle: $c^2>a^2+b^2$, $c^2=a^2+b^2$, or $c^2<a^2+b^2$?

    $c^2=a^2+b^2$. The Pythagorean theorem guarantees equality for right triangles.

51. Find $c$ if a right triangle has legs $a=5$ and $b=12$.

    $c=13$. Apply theorem: $5^2+12^2=25+144=169$, so $c=sqrt{169}=13$.

52. What is the converse of the Pythagorean Theorem (in terms of $a,b,c$ with $c$ largest)?

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

53. In the rearrangement proof, what is the area expression for the central square with side $c$?

    $c^2$. The inner square has side length equal to the hypotenuse.

54. Identify the triangle type if $a^2+b^2<c^2$ for sides $a,b,c$ with $c$ longest.

    Obtuse triangle. When leg squares are less than hypotenuse squared, one angle exceeds 90°.

55. Which side is the hypotenuse in a right triangle when using $a^2+b^2=c^2$?

    $c$ is the hypotenuse (the longest side, opposite the right angle). The hypotenuse is always the longest side in a right triangle.

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