Properties of Right Triangles - PSAT Math
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In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is the hypotenuse in terms of a leg $x$?
In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is the hypotenuse in terms of a leg $x$?
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$x\sqrt{2}$. In isosceles right triangles, multiply leg by $\sqrt{2}$ to get hypotenuse.
$x\sqrt{2}$. In isosceles right triangles, multiply leg by $\sqrt{2}$ to get hypotenuse.
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What is the converse of the Pythagorean Theorem for side lengths $a,b,c$ with $c$ largest?
What is the converse of the Pythagorean Theorem for side lengths $a,b,c$ with $c$ largest?
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If $a^2+b^2=c^2$, the triangle is right. If the equation holds, then the triangle must have a right angle.
If $a^2+b^2=c^2$, the triangle is right. If the equation holds, then the triangle must have a right angle.
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Identify the hypotenuse in a right triangle in terms of its relationship to the right angle.
Identify the hypotenuse in a right triangle in terms of its relationship to the right angle.
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The side opposite the $90^\circ$ angle. The hypotenuse is always opposite the right angle in any right triangle.
The side opposite the $90^\circ$ angle. The hypotenuse is always opposite the right angle in any right triangle.
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State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle in simplest form.
State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle in simplest form.
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$1:1:\sqrt{2}$. Isosceles right triangles have two equal legs and hypotenuse $\sqrt{2}$ times a leg.
$1:1:\sqrt{2}$. Isosceles right triangles have two equal legs and hypotenuse $\sqrt{2}$ times a leg.
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State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle in simplest form.
State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle in simplest form.
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$1:\sqrt{3}:2$. Short leg opposite $30°$, long leg opposite $60°$, hypotenuse opposite $90°$.
$1:\sqrt{3}:2$. Short leg opposite $30°$, long leg opposite $60°$, hypotenuse opposite $90°$.
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State the area formula for a right triangle with legs $a$ and $b$ as base and height.
State the area formula for a right triangle with legs $a$ and $b$ as base and height.
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$A=\frac{1}{2}ab$. The legs form the base and height, so area is half their product.
$A=\frac{1}{2}ab$. The legs form the base and height, so area is half their product.
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State the definition of a Pythagorean triple in terms of integers $a,b,c$ with $c$ largest.
State the definition of a Pythagorean triple in terms of integers $a,b,c$ with $c$ largest.
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Integers with $a^2+b^2=c^2$. Three positive integers satisfying the Pythagorean theorem.
Integers with $a^2+b^2=c^2$. Three positive integers satisfying the Pythagorean theorem.
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Find the hypotenuse $c$ if the legs are $6$ and $8$.
Find the hypotenuse $c$ if the legs are $6$ and $8$.
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$10$. Apply Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.
$10$. Apply Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.
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Find the missing leg $a$ if $c=13$ and $b=5$ in a right triangle.
Find the missing leg $a$ if $c=13$ and $b=5$ in a right triangle.
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$12$. Rearrange Pythagorean theorem: $a=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.
$12$. Rearrange Pythagorean theorem: $a=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.
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Identify whether side lengths $9,12,15$ form a right triangle.
Identify whether side lengths $9,12,15$ form a right triangle.
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Yes, because $9^2+12^2=15^2$. Check if $9^2+12^2=15^2$: $81+144=225$, which is true.
Yes, because $9^2+12^2=15^2$. Check if $9^2+12^2=15^2$: $81+144=225$, which is true.
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Find the long leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $7$.
Find the long leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $7$.
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$7\sqrt{3}$. Multiply short leg by $\sqrt{3}$ to get long leg in $30°$-$60°$-$90°$ triangle.
$7\sqrt{3}$. Multiply short leg by $\sqrt{3}$ to get long leg in $30°$-$60°$-$90°$ triangle.
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Find the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $9$.
Find the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $9$.
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$9\sqrt{2}$. In $45°$-$45°$-$90°$ triangle, hypotenuse equals leg times $\sqrt{2}$.
$9\sqrt{2}$. In $45°$-$45°$-$90°$ triangle, hypotenuse equals leg times $\sqrt{2}$.
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Find the area of a right triangle with legs $10$ and $12$.
Find the area of a right triangle with legs $10$ and $12$.
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$60$. Use $A=\frac{1}{2}ab=\frac{1}{2}(10)(12)=60$.
$60$. Use $A=\frac{1}{2}ab=\frac{1}{2}(10)(12)=60$.
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Find the distance between points $(1,2)$ and $(7,10)$.
Find the distance between points $(1,2)$ and $(7,10)$.
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$10$. $d=\sqrt{(7-1)^2+(10-2)^2}=\sqrt{36+64}=\sqrt{100}=10$.
$10$. $d=\sqrt{(7-1)^2+(10-2)^2}=\sqrt{36+64}=\sqrt{100}=10$.
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Find the slope of a line perpendicular to a line with slope $\frac{2}{3}$.
Find the slope of a line perpendicular to a line with slope $\frac{2}{3}$.
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$-\frac{3}{2}$. Take negative reciprocal: $-\frac{1}{\frac{2}{3}}=-\frac{3}{2}$.
$-\frac{3}{2}$. Take negative reciprocal: $-\frac{1}{\frac{2}{3}}=-\frac{3}{2}$.
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Identify the missing leg: right triangle with hypotenuse $13$ and leg $5$; what is the other leg?
Identify the missing leg: right triangle with hypotenuse $13$ and leg $5$; what is the other leg?
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$12$. Use $a^2 + b^2 = c^2$: $5^2 + b^2 = 13^2$, so $b = \sqrt{144} = 12$.
$12$. Use $a^2 + b^2 = c^2$: $5^2 + b^2 = 13^2$, so $b = \sqrt{144} = 12$.
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State the side-length ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle.
State the side-length ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle.
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$1:1:\sqrt{2}$. Legs are equal, hypotenuse is $\sqrt{2}$ times a leg.
$1:1:\sqrt{2}$. Legs are equal, hypotenuse is $\sqrt{2}$ times a leg.
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What is the converse of the Pythagorean Theorem for positive side lengths $a\le b<c$?
What is the converse of the Pythagorean Theorem for positive side lengths $a\le b<c$?
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If $a^2+b^2=c^2$, the triangle is right. Tests if a triangle is right-angled using side lengths.
If $a^2+b^2=c^2$, the triangle is right. Tests if a triangle is right-angled using side lengths.
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Classify a triangle with sides $6,8,10$ using the Pythagorean Theorem.
Classify a triangle with sides $6,8,10$ using the Pythagorean Theorem.
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Right triangle. $6^2 + 8^2 = 36 + 64 = 100 = 10^2$, so it's right.
Right triangle. $6^2 + 8^2 = 36 + 64 = 100 = 10^2$, so it's right.
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Find the missing leg length if the hypotenuse is $13$ and one leg is $5$.
Find the missing leg length if the hypotenuse is $13$ and one leg is $5$.
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$12$. Use $a^2+b^2=c^2$: $5^2+b^2=13^2$, so $b^2=169-25=144$, thus $b=12$.
$12$. Use $a^2+b^2=c^2$: $5^2+b^2=13^2$, so $b^2=169-25=144$, thus $b=12$.
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What is the length of the hypotenuse if the legs are $6$ and $8$?
What is the length of the hypotenuse if the legs are $6$ and $8$?
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$10$. Apply Pythagorean theorem: $\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.
$10$. Apply Pythagorean theorem: $\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.
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What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ derived from a right triangle?
What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ derived from a right triangle?
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$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses the Pythagorean theorem on the coordinate differences.
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses the Pythagorean theorem on the coordinate differences.
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What are the side lengths in the Pythagorean triple with legs $5$ and $12$?
What are the side lengths in the Pythagorean triple with legs $5$ and $12$?
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$5,12,13$. The hypotenuse is $\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$.
$5,12,13$. The hypotenuse is $\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$.
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What are the side lengths in the Pythagorean triple with legs $8$ and $15$?
What are the side lengths in the Pythagorean triple with legs $8$ and $15$?
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$8,15,17$. The hypotenuse is $\sqrt{8^2+15^2}=\sqrt{64+225}=\sqrt{289}=17$.
$8,15,17$. The hypotenuse is $\sqrt{8^2+15^2}=\sqrt{64+225}=\sqrt{289}=17$.
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Find the long leg of a $30^\circ-60^\circ-90^\circ$ triangle if the short leg is $4$.
Find the long leg of a $30^\circ-60^\circ-90^\circ$ triangle if the short leg is $4$.
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$4\sqrt{3}$. Long leg = short leg × $\sqrt{3}$ in 30-60-90 triangle: $4\times\sqrt{3}$.
$4\sqrt{3}$. Long leg = short leg × $\sqrt{3}$ in 30-60-90 triangle: $4\times\sqrt{3}$.
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State the side ratio for a $45^\circ-45^\circ-90^\circ$ right triangle in simplest radical form.
State the side ratio for a $45^\circ-45^\circ-90^\circ$ right triangle in simplest radical form.
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$1:1:\sqrt{2}$. Equal legs to hypotenuse ratio in an isosceles right triangle.
$1:1:\sqrt{2}$. Equal legs to hypotenuse ratio in an isosceles right triangle.
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In a $45^\circ-45^\circ-90^\circ$ triangle, what is the hypotenuse if a leg is $x$?
In a $45^\circ-45^\circ-90^\circ$ triangle, what is the hypotenuse if a leg is $x$?
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$x\sqrt{2}$. In a 45-45-90 triangle, hypotenuse = leg × $\sqrt{2}$.
$x\sqrt{2}$. In a 45-45-90 triangle, hypotenuse = leg × $\sqrt{2}$.
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In a $45^\circ-45^\circ-90^\circ$ triangle, what is each leg if the hypotenuse is $h$?
In a $45^\circ-45^\circ-90^\circ$ triangle, what is each leg if the hypotenuse is $h$?
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$\frac{h}{\sqrt{2}}$. Divide hypotenuse by $\sqrt{2}$ to find each equal leg.
$\frac{h}{\sqrt{2}}$. Divide hypotenuse by $\sqrt{2}$ to find each equal leg.
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State the side ratio for a $30^\circ-60^\circ-90^\circ$ right triangle in simplest radical form.
State the side ratio for a $30^\circ-60^\circ-90^\circ$ right triangle in simplest radical form.
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$1:\sqrt{3}:2$. Short leg : long leg : hypotenuse ratio for 30-60-90 triangles.
$1:\sqrt{3}:2$. Short leg : long leg : hypotenuse ratio for 30-60-90 triangles.
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In a $30^\circ-60^\circ-90^\circ$ triangle, what is the hypotenuse if the short leg is $x$?
In a $30^\circ-60^\circ-90^\circ$ triangle, what is the hypotenuse if the short leg is $x$?
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$2x$. Hypotenuse is twice the short leg (opposite 30°) in 30-60-90 triangles.
$2x$. Hypotenuse is twice the short leg (opposite 30°) in 30-60-90 triangles.
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