Apply Pythagorean Theorem to Problems - 8th Grade Math
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Find the length of the diagonal of a $7$ by $24$ rectangle.
Find the length of the diagonal of a $7$ by $24$ rectangle.
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$25$. $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$.
$25$. $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$.
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Identify which side is the hypotenuse in a right triangle when using $a^2 + b^2 = c^2$.
Identify which side is the hypotenuse in a right triangle when using $a^2 + b^2 = c^2$.
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The side opposite the $90^\circ$ angle, labeled $c$. The hypotenuse is always the longest side, opposite the right angle.
The side opposite the $90^\circ$ angle, labeled $c$. The hypotenuse is always the longest side, opposite the right angle.
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State the Pythagorean Theorem formula for a right triangle with legs $a$, $b$ and hypotenuse $c$.
State the Pythagorean Theorem formula for a right triangle with legs $a$, $b$ and hypotenuse $c$.
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$a^2 + b^2 = c^2$. The sum of the squares of the legs equals the square of the hypotenuse.
$a^2 + b^2 = c^2$. The sum of the squares of the legs equals the square of the hypotenuse.
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State the formula to find a missing leg $a$ when the hypotenuse is $c$ and the other leg is $b$.
State the formula to find a missing leg $a$ when the hypotenuse is $c$ and the other leg is $b$.
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$a = \sqrt{c^2 - b^2}$. Rearrange $a^2 + b^2 = c^2$ to isolate $a^2$, then take the square root.
$a = \sqrt{c^2 - b^2}$. Rearrange $a^2 + b^2 = c^2$ to isolate $a^2$, then take the square root.
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State the formula to find the hypotenuse $c$ when the legs are $a$ and $b$.
State the formula to find the hypotenuse $c$ when the legs are $a$ and $b$.
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$c = \sqrt{a^2 + b^2}$. Take the square root of the sum of the squared legs.
$c = \sqrt{a^2 + b^2}$. Take the square root of the sum of the squared legs.
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Find the distance between points $(2,3)$ and $(5,7)$ using the Pythagorean Theorem.
Find the distance between points $(2,3)$ and $(5,7)$ using the Pythagorean Theorem.
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$5$. $\sqrt{(5-2)^2 + (7-3)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
$5$. $\sqrt{(5-2)^2 + (7-3)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
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Find the hypotenuse $c$ when the legs are $a = 3$ and $b = 4$.
Find the hypotenuse $c$ when the legs are $a = 3$ and $b = 4$.
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$c = 5$. $3^2 + 4^2 = 9 + 16 = 25$, so $c = \sqrt{25} = 5$.
$c = 5$. $3^2 + 4^2 = 9 + 16 = 25$, so $c = \sqrt{25} = 5$.
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A ladder is $10$ ft long and reaches $6$ ft high. What is the distance from the wall to the ladder base?
A ladder is $10$ ft long and reaches $6$ ft high. What is the distance from the wall to the ladder base?
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$8$ ft. Base distance: $\sqrt{10^2 - 6^2} = \sqrt{100 - 36} = 8$ ft.
$8$ ft. Base distance: $\sqrt{10^2 - 6^2} = \sqrt{100 - 36} = 8$ ft.
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A right triangle has legs $7$ and $24$. What is the hypotenuse?
A right triangle has legs $7$ and $24$. What is the hypotenuse?
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$25$. $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$.
$25$. $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$.
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Find the missing leg $b$ when $c = 13$ and $a = 5$.
Find the missing leg $b$ when $c = 13$ and $a = 5$.
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$b = 12$. $b^2 = 13^2 - 5^2 = 169 - 25 = 144$, so $b = 12$.
$b = 12$. $b^2 = 13^2 - 5^2 = 169 - 25 = 144$, so $b = 12$.
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Find the missing leg $a$ when $c = 10$ and $b = 6$.
Find the missing leg $a$ when $c = 10$ and $b = 6$.
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$a = 8$. $a^2 = 10^2 - 6^2 = 100 - 36 = 64$, so $a = 8$.
$a = 8$. $a^2 = 10^2 - 6^2 = 100 - 36 = 64$, so $a = 8$.
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Find the hypotenuse $c$ when the legs are $a = 5$ and $b = 12$.
Find the hypotenuse $c$ when the legs are $a = 5$ and $b = 12$.
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$c = 13$. $5^2 + 12^2 = 25 + 144 = 169$, so $c = \sqrt{169} = 13$.
$c = 13$. $5^2 + 12^2 = 25 + 144 = 169$, so $c = \sqrt{169} = 13$.
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Find the length of the hypotenuse when the legs are $a = 8$ and $b = 15$.
Find the length of the hypotenuse when the legs are $a = 8$ and $b = 15$.
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$17$. $\sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$.
$17$. $\sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$.
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Find the diagonal of a $6$ by $8$ rectangle using the Pythagorean Theorem.
Find the diagonal of a $6$ by $8$ rectangle using the Pythagorean Theorem.
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$10$. Diagonal forms hypotenuse: $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = 10$.
$10$. Diagonal forms hypotenuse: $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = 10$.
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A rectangle has diagonal $13$ and one side $5$. Find the other side length.
A rectangle has diagonal $13$ and one side $5$. Find the other side length.
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$12$. $\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$.
$12$. $\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$.
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Identify whether the side lengths $5$, $6$, and $7$ form a right triangle using $a^2+b^2=c^2$.
Identify whether the side lengths $5$, $6$, and $7$ form a right triangle using $a^2+b^2=c^2$.
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No, because $5^2 + 6^2 \ne 7^2$. Check: $25 + 36 = 61 \ne 49$, so not a right triangle.
No, because $5^2 + 6^2 \ne 7^2$. Check: $25 + 36 = 61 \ne 49$, so not a right triangle.
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Find the distance between points $(0,0)$ and $(6,8)$ using the Pythagorean Theorem.
Find the distance between points $(0,0)$ and $(6,8)$ using the Pythagorean Theorem.
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$10$. Distance formula uses Pythagorean theorem: $\sqrt{6^2 + 8^2} = 10$.
$10$. Distance formula uses Pythagorean theorem: $\sqrt{6^2 + 8^2} = 10$.
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Identify whether the side lengths $6$, $8$, and $10$ form a right triangle using $a^2+b^2=c^2$.
Identify whether the side lengths $6$, $8$, and $10$ form a right triangle using $a^2+b^2=c^2$.
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Yes, because $6^2 + 8^2 = 10^2$. Check: $36 + 64 = 100$, which equals $10^2$.
Yes, because $6^2 + 8^2 = 10^2$. Check: $36 + 64 = 100$, which equals $10^2$.
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Find the diagonal of a square with side length $9$ using the Pythagorean Theorem.
Find the diagonal of a square with side length $9$ using the Pythagorean Theorem.
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$9\sqrt{2}$. Square diagonal: $\sqrt{9^2 + 9^2} = \sqrt{162} = 9\sqrt{2}$.
$9\sqrt{2}$. Square diagonal: $\sqrt{9^2 + 9^2} = \sqrt{162} = 9\sqrt{2}$.
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Find the hypotenuse when the legs are $5$ and $12$.
Find the hypotenuse when the legs are $5$ and $12$.
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$13$. $\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
$13$. $\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
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Find the hypotenuse when the legs are $9$ and $12$.
Find the hypotenuse when the legs are $9$ and $12$.
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$15$. $\sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$.
$15$. $\sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$.
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What is the formula for the hypotenuse $c$ in terms of legs $a$ and $b$?
What is the formula for the hypotenuse $c$ in terms of legs $a$ and $b$?
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$c = \sqrt{a^2 + b^2}$. Rearrange the Pythagorean theorem to solve for the hypotenuse.
$c = \sqrt{a^2 + b^2}$. Rearrange the Pythagorean theorem to solve for the hypotenuse.
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Find the missing leg when $c=13$ and one leg is $5$.
Find the missing leg when $c=13$ and one leg is $5$.
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$12$. $\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$.
$12$. $\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$.
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Find the hypotenuse when the legs are $6$ and $8$.
Find the hypotenuse when the legs are $6$ and $8$.
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$10$. $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.
$10$. $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.
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What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ from the Pythagorean Theorem?
What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ from the Pythagorean Theorem?
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$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Treats horizontal and vertical distances as legs of a right triangle.
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Treats horizontal and vertical distances as legs of a right triangle.
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