8th Grade Math Flashcards: Explain Pythagorean Theorem Proof

What this deck covers

This deck focuses on Explain Pythagorean Theorem Proof, giving you a quick way to review the definitions, rules, and examples that matter most for 8th Grade Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: In a rearrangement proof, what is the area of a right triangle with legs $a$ and $b$?

Answer: rac{1}{2}ab. Uses the standard triangle area formula: base times height divided by 2.

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

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

Flashcard 3: Use the converse: Are side lengths $6,8,10$ a right triangle (yes or no)?

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

Flashcard 4: Find the missing leg $a$ if $c=10$ and $b=6$ in a right triangle.

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

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

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

Flashcard 6: Find $c$ if a right triangle has legs $a=3$ and $b=4$.

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

Flashcard 7: What is the area expression for the large square of side $a+b$ in the rearrangement proof?

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

Flashcard 8: What is the converse of the Pythagorean Theorem (in terms of $a,b,c$ with $c$ largest)?

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

Flashcard 9: Which statement is correct for a right triangle: $a^2+b^2=c^2$ or $a+b=c$?

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

Flashcard 10: What conclusion is justified if $a^2+b^2<c^2$ for sides $a,b,c$ with $c$ largest?

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

Flashcard 11: What conclusion is justified if $a^2+b^2>c^2$ for sides $a,b,c$ with $c$ largest?

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

Flashcard 12: Which side must be used as $c$ when applying $a^2+b^2=c^2$ to a triangle?

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

Flashcard 13: Which equation tests whether a triangle with sides $a,b,c$ (with $c$ largest) is right?

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

Flashcard 14: What is the area of a square with side length $s$ (used in some Pythagorean proofs)?

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

Flashcard 15: In the common rearrangement proof, what is the side length of the large square built from legs $a$ and $b$?

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

Flashcard 16: Use the converse: Are side lengths $4,5,6$ a right triangle (yes or no)?

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

Flashcard 17: Identify the error: A student used $7^2+24^2=25^2$ but labeled $7$ as the hypotenuse. What is the correction?

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

Flashcard 18: In the rearrangement proof, what is the area expression for the central square with side $c$?

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

Flashcard 19: What is the Pythagorean Theorem for a right triangle with legs $a,b$ and hypotenuse $c$?

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

Flashcard 20: Identify the first step to use the converse: which side should be chosen as $c$?

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

Flashcard 21: What is the area of a right triangle with legs $a$ and $b$?

Answer: rac{1}{2}ab. Uses the formula: base times height divided by 2.

Flashcard 22: What does the area model proof compare to show $a^2+b^2=c^2$?

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

Flashcard 23: What does the symbol $c$ represent in $a^2+b^2=c^2$ for a right triangle?

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

Flashcard 24: What is the converse of the Pythagorean Theorem using side lengths $a,b,c$ with $c$ longest?

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

Flashcard 25: What identity is used to expand $(a+b)^2$ in the area proof?

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

Flashcard 26: After expanding $(a+b)^2=a^2+2ab+b^2$ and setting it equal to $2ab+c^2$, what conclusion follows?

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

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

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

Flashcard 28: In the square proof, what is the area of the central square if its side length is $c$?

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

Flashcard 29: In the square proof, what is the total area of the four congruent right triangles with legs $a$ and $b$?

Answer: 4ig(rac{1}{2}abig)=2ab. Four triangles each with area rac{1}{2}ab sum to $2ab$.

Flashcard 30: In the square proof, what is the area of the large square with side length $a+b$?

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