Explain Pythagorean Theorem Proof - 8th Grade Math
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In a rearrangement proof, what is the area of a right triangle with legs $a$ and $b$?
In a rearrangement proof, what is the area of a right triangle with legs $a$ and $b$?
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$ rac{1}{2}ab$. Uses the standard triangle area formula: base times height divided by 2.
$ rac{1}{2}ab$. Uses the standard triangle area formula: base times height divided by 2.
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What does the Pythagorean Theorem say about the side labeled $c$ in $a^2+b^2=c^2$?
What does the Pythagorean Theorem say about the side labeled $c$ in $a^2+b^2=c^2$?
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$c$ is the hypotenuse (side opposite the right angle). In a right triangle, $c$ must be the longest side.
$c$ is the hypotenuse (side opposite the right angle). In a right triangle, $c$ must be the longest side.
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Use the converse: Are side lengths $6,8,10$ a right triangle (yes or no)?
Use the converse: Are side lengths $6,8,10$ a right triangle (yes or no)?
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Yes, because $6^2+8^2=10^2$. Check: $36+64=100$ is true, confirming a right triangle.
Yes, because $6^2+8^2=10^2$. Check: $36+64=100$ is true, confirming a right triangle.
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Find the missing leg $a$ if $c=10$ and $b=6$ in a right triangle.
Find the missing leg $a$ if $c=10$ and $b=6$ in a right triangle.
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$a=8$. Rearrange: $a^2=c^2-b^2=100-36=64$, so $a=8$.
$a=8$. Rearrange: $a^2=c^2-b^2=100-36=64$, so $a=8$.
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Find $c$ if a right triangle has legs $a=5$ and $b=12$.
Find $c$ if a right triangle has legs $a=5$ and $b=12$.
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$c=13$. Apply theorem: $5^2+12^2=25+144=169$, so $c=sqrt{169}=13$.
$c=13$. Apply theorem: $5^2+12^2=25+144=169$, so $c=sqrt{169}=13$.
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Find $c$ if a right triangle has legs $a=3$ and $b=4$.
Find $c$ if a right triangle has legs $a=3$ and $b=4$.
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$c=5$. Apply theorem: $3^2+4^2=9+16=25$, so $c=sqrt{25}=5$.
$c=5$. Apply theorem: $3^2+4^2=9+16=25$, so $c=sqrt{25}=5$.
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What is the area expression for the large square of side $a+b$ in the rearrangement proof?
What is the area expression for the large square of side $a+b$ in the rearrangement proof?
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$(a+b)^2$. Expands to $a^2+2ab+b^2$ using the binomial formula.
$(a+b)^2$. Expands to $a^2+2ab+b^2$ using the binomial formula.
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What is the converse of the Pythagorean Theorem (in terms of $a,b,c$ with $c$ largest)?
What is the converse of the Pythagorean Theorem (in terms of $a,b,c$ with $c$ largest)?
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If $a^2+b^2=c^2$, then the triangle is right. If the equation holds, then the angle opposite side $c$ is 90°.
If $a^2+b^2=c^2$, then the triangle is right. If the equation holds, then the angle opposite side $c$ is 90°.
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Which statement is correct for a right triangle: $a^2+b^2=c^2$ or $a+b=c$?
Which statement is correct for a right triangle: $a^2+b^2=c^2$ or $a+b=c$?
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$a^2+b^2=c^2$. The Pythagorean theorem uses squares, not simple addition.
$a^2+b^2=c^2$. The Pythagorean theorem uses squares, not simple addition.
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What conclusion is justified if $a^2+b^2<c^2$ for sides $a,b,c$ with $c$ largest?
What conclusion is justified if $a^2+b^2<c^2$ for sides $a,b,c$ with $c$ largest?
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The triangle is obtuse. When the sum of leg squares is less than hypotenuse squared, angle exceeds 90°.
The triangle is obtuse. When the sum of leg squares is less than hypotenuse squared, angle exceeds 90°.
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What conclusion is justified if $a^2+b^2>c^2$ for sides $a,b,c$ with $c$ largest?
What conclusion is justified if $a^2+b^2>c^2$ for sides $a,b,c$ with $c$ largest?
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The triangle is acute. When the sum of leg squares exceeds hypotenuse squared, all angles are under 90°.
The triangle is acute. When the sum of leg squares exceeds hypotenuse squared, all angles are under 90°.
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Which side must be used as $c$ when applying $a^2+b^2=c^2$ to a triangle?
Which side must be used as $c$ when applying $a^2+b^2=c^2$ to a triangle?
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The longest side (the hypotenuse if the triangle is right). The theorem requires $c$ to be the hypotenuse for validity.
The longest side (the hypotenuse if the triangle is right). The theorem requires $c$ to be the hypotenuse for validity.
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Which equation tests whether a triangle with sides $a,b,c$ (with $c$ largest) is right?
Which equation tests whether a triangle with sides $a,b,c$ (with $c$ largest) is right?
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Check whether $a^2+b^2=c^2$. The converse tests if a triangle is right by checking this equality.
Check whether $a^2+b^2=c^2$. The converse tests if a triangle is right by checking this equality.
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What is the area of a square with side length $s$ (used in some Pythagorean proofs)?
What is the area of a square with side length $s$ (used in some Pythagorean proofs)?
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$s^2$. Area equals side length squared for any square.
$s^2$. Area equals side length squared for any square.
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In the common rearrangement proof, what is the side length of the large square built from legs $a$ and $b$?
In the common rearrangement proof, what is the side length of the large square built from legs $a$ and $b$?
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$a+b$. The outer square has sides equal to the sum of the two legs.
$a+b$. The outer square has sides equal to the sum of the two legs.
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Use the converse: Are side lengths $4,5,6$ a right triangle (yes or no)?
Use the converse: Are side lengths $4,5,6$ a right triangle (yes or no)?
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No, because $4^2+5^2\ne6^2$. Check: $16+25=41
eq 36$, so not a right triangle.
No, because $4^2+5^2\ne6^2$. Check: $16+25=41 eq 36$, so not a right triangle.
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Identify the error: A student used $7^2+24^2=25^2$ but labeled $7$ as the hypotenuse. What is the correction?
Identify the error: A student used $7^2+24^2=25^2$ but labeled $7$ as the hypotenuse. What is the correction?
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The hypotenuse is $25$ (the longest side), not $7$. In the theorem, $c$ must be the longest side to be the hypotenuse.
The hypotenuse is $25$ (the longest side), not $7$. In the theorem, $c$ must be the longest side to be the hypotenuse.
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In the rearrangement proof, what is the area expression for the central square with side $c$?
In the rearrangement proof, what is the area expression for the central square with side $c$?
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$c^2$. The inner square has side length equal to the hypotenuse.
$c^2$. The inner square has side length equal to the hypotenuse.
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What is the Pythagorean Theorem for a right triangle with legs $a,b$ and hypotenuse $c$?
What is the Pythagorean Theorem for a right triangle with legs $a,b$ and hypotenuse $c$?
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$a^2+b^2=c^2$. States that the sum of squares of the legs equals the square of the hypotenuse.
$a^2+b^2=c^2$. States that the sum of squares of the legs equals the square of the hypotenuse.
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Identify the first step to use the converse: which side should be chosen as $c$?
Identify the first step to use the converse: which side should be chosen as $c$?
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Choose the longest side as $c$. The converse requires comparing with the longest side squared.
Choose the longest side as $c$. The converse requires comparing with the longest side squared.
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What is the area of a right triangle with legs $a$ and $b$?
What is the area of a right triangle with legs $a$ and $b$?
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$ rac{1}{2}ab$. Uses the formula: base times height divided by 2.
$ rac{1}{2}ab$. Uses the formula: base times height divided by 2.
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What does the area model proof compare to show $a^2+b^2=c^2$?
What does the area model proof compare to show $a^2+b^2=c^2$?
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Two expressions for the same square’s area. The proof shows $(a+b)^2 = 4$ triangles $+ c^2$ square.
Two expressions for the same square’s area. The proof shows $(a+b)^2 = 4$ triangles $+ c^2$ square.
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What does the symbol $c$ represent in $a^2+b^2=c^2$ for a right triangle?
What does the symbol $c$ represent in $a^2+b^2=c^2$ for a right triangle?
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$c$ is the hypotenuse (longest side). In right triangles, the hypotenuse is always opposite the right angle.
$c$ is the hypotenuse (longest side). In right triangles, the hypotenuse is always opposite the right angle.
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What is the converse of the Pythagorean Theorem using side lengths $a,b,c$ with $c$ longest?
What is the converse of the Pythagorean Theorem using side lengths $a,b,c$ with $c$ longest?
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If $a^2+b^2=c^2$, then the triangle is right. Reverses the theorem: if the equation holds, then the angle is 90°.
If $a^2+b^2=c^2$, then the triangle is right. Reverses the theorem: if the equation holds, then the angle is 90°.
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What identity is used to expand $(a+b)^2$ in the area proof?
What identity is used to expand $(a+b)^2$ in the area proof?
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$(a+b)^2=a^2+2ab+b^2$. Standard binomial expansion: square of sum equals sum of squares plus twice the product.
$(a+b)^2=a^2+2ab+b^2$. Standard binomial expansion: square of sum equals sum of squares plus twice the product.
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After expanding $(a+b)^2=a^2+2ab+b^2$ and setting it equal to $2ab+c^2$, what conclusion follows?
After expanding $(a+b)^2=a^2+2ab+b^2$ and setting it equal to $2ab+c^2$, what conclusion follows?
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$a^2+b^2=c^2$. Cancel $2ab$ from both sides to get the Pythagorean theorem.
$a^2+b^2=c^2$. Cancel $2ab$ from both sides to get the Pythagorean theorem.
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Identify whether sides $3,4,5$ form a right triangle using the converse.
Identify whether sides $3,4,5$ form a right triangle using the converse.
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Yes, because $3^2+4^2=5^2$. Check: $9+16=25$ ✓, so the triangle has a right angle.
Yes, because $3^2+4^2=5^2$. Check: $9+16=25$ ✓, so the triangle has a right angle.
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In the square proof, what is the area of the central square if its side length is $c$?
In the square proof, what is the area of the central square if its side length is $c$?
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$c^2$. The hypotenuse forms the side of the inner square.
$c^2$. The hypotenuse forms the side of the inner square.
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In the square proof, what is the total area of the four congruent right triangles with legs $a$ and $b$?
In the square proof, what is the total area of the four congruent right triangles with legs $a$ and $b$?
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$4ig( rac{1}{2}abig)=2ab$. Four triangles each with area $ rac{1}{2}ab$ sum to $2ab$.
$4ig( rac{1}{2}abig)=2ab$. Four triangles each with area $ rac{1}{2}ab$ sum to $2ab$.
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In the square proof, what is the area of the large square with side length $a+b$?
In the square proof, what is the area of the large square with side length $a+b$?
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$(a+b)^2$. The large square has side length equal to the sum of the two legs.
$(a+b)^2$. The large square has side length equal to the sum of the two legs.
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