PSAT Math Flashcards: Properties Of Right Triangles

What this deck covers

This deck focuses on Properties Of Right Triangles, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT 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: State the side-length ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$.

Answer: Long leg $x\sqrt{3}$; hypotenuse $2x$. Special triangle with sides in ratio $1:\sqrt{3}:2$.

Flashcard 2: What is the relationship between the two acute angles in any right triangle?

Answer: They are complementary: $\theta_1+\theta_2=90^\circ$. The two non-right angles must sum to $90°$ since triangle angles sum to $180°$.

Flashcard 3: Find the short leg of a $30^\circ-60^\circ-90^\circ$ triangle if the hypotenuse is $18$.

Answer: $9$. Short leg = hypotenuse ÷ 2 in 30-60-90 triangle: $18÷2=9$.

Flashcard 4: Find the distance between points $(1,2)$ and $(5,5)$ using a right-triangle interpretation.

Answer: $5$. Distance = $\sqrt{(5-1)^2+(5-2)^2}=\sqrt{16+9}=\sqrt{25}=5$.

Flashcard 5: In a $30^\circ-60^\circ-90^\circ$ triangle, what is the hypotenuse if the short leg is $x$?

Answer: $2x$. Hypotenuse is twice the short leg (opposite 30°) in 30-60-90 triangles.

Flashcard 6: In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the hypotenuse if the short leg is $9$?

Answer: $18$. Double the short leg for 30-60-90 hypotenuse: $2 × 9 = 18$.

Flashcard 7: Find the short leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle with hypotenuse $18$.

Answer: $9$. Short leg is half the hypotenuse in a 30-60-90 triangle.

Flashcard 8: Find the distance between points $(1,2)$ and $(7,10)$.

Answer: $10$. $d=\sqrt{(7-1)^2+(10-2)^2}=\sqrt{36+64}=\sqrt{100}=10$.

Flashcard 9: In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, if the short leg is $7$, what is the hypotenuse?

Answer: $14$. Hypotenuse is twice the short leg in $30°$-$60°$-$90°$ triangles.

Flashcard 10: State the slope formula using points $(x_1,y_1)$ and $(x_2,y_2)$ (rise over run).

Answer: $\frac{y_2-y_1}{x_2-x_1}$. Vertical change divided by horizontal change gives the slope.

Flashcard 11: Find the hypotenuse $c$ if the legs are $6$ and $8$.

Answer: $10$. Using Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.

Flashcard 12: In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, find the hypotenuse if a leg is $7$.

Answer: $7\sqrt{2}$. Multiply leg by $\sqrt{2}$ for hypotenuse in 45-45-90 triangle.

Flashcard 13: Identify the acute angle measure if the other acute angle in a right triangle is $35^\circ$.

Answer: $55^\circ$. Acute angles are complementary: $90°-35°=55°$.

Flashcard 14: In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, if the hypotenuse is $18$, what is the short leg?

Answer: $9$. Short leg is half the hypotenuse in $30°$-$60°$-$90°$ triangles.

Flashcard 15: Find the long leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $5$.

Answer: $5\sqrt{3}$. In a $30°$-$60°$-$90°$ triangle, long leg = short leg × $\sqrt{3}$.

Flashcard 16: Find the missing leg: right triangle with hypotenuse $25$ and leg $7$; what is the other leg?

Answer: $24$. Solve $7^2 + b^2 = 25^2$ to get $b = \sqrt{576} = 24$.

Flashcard 17: What is the definition of the legs in a right triangle?

Answer: The two sides that form the $90^\circ$ angle. These perpendicular sides meet at the right angle.

Flashcard 18: Find the area of a right triangle with legs $9$ and $4$.

Answer: $18$. Area $=\frac{1}{2}(9)(4)=\frac{36}{2}=18$.

Flashcard 19: State the formula for the area of a right triangle with legs $a$ and $b$.

Answer: $A=\frac{1}{2}ab$. Half the product of the two legs (perpendicular sides).

Flashcard 20: Identify the hypotenuse in a right triangle in relation to the right angle.

Answer: The side opposite the $90^\circ$ angle. The longest side in a right triangle, always opposite the right angle.

Flashcard 21: What is the tangent ratio in a right triangle for acute angle $\theta$?

Answer: $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Ratio of the side opposite to the side adjacent to angle $\theta$.

Flashcard 22: State the special right triangle ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle.

Answer: Side ratio $1:\sqrt{3}:2$. Shortest side opposite $30°$, longest opposite $90°$, middle opposite $60°$.

Flashcard 23: In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, if a leg is $9$, what is the hypotenuse?

Answer: $9\sqrt{2}$. Multiply leg by $\sqrt{2}$ in $45°$-$45°$-$90°$ triangles.

Flashcard 24: State the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ using a right-triangle interpretation.

Answer: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses Pythagorean theorem on the coordinate differences.

Flashcard 25: What is the converse of the Pythagorean Theorem for positive side lengths $a\le b<c$?

Answer: If $a^2+b^2=c^2$, the triangle is right. Tests if a triangle is right-angled using side lengths.

Flashcard 26: Identify the hypotenuse in a right triangle in relation to the $90^\circ$ angle.

Answer: The side opposite the $90^\circ$ angle. The longest side in a right triangle, opposite the right angle.

Flashcard 27: What is the formula for the hypotenuse $c$ in terms of legs $a$ and $b$?

Answer: $c=\sqrt{a^2+b^2}$. Solve the Pythagorean theorem for the hypotenuse.

Flashcard 28: In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the long leg in terms of short leg $x$?

Answer: $x\sqrt{3}$. The side opposite $60°$ is $\sqrt{3}$ times the side opposite $30°$.

Flashcard 29: What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ derived from right triangles?

Answer: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses the Pythagorean theorem on the coordinate plane.

Flashcard 30: What is the formula for a leg $a$ in terms of hypotenuse $c$ and other leg $b$?

Answer: $a=\sqrt{c^2-b^2}$. Rearrange $a^2+b^2=c^2$ to isolate the leg $a$.

Flashcard 31: What is the slope of a horizontal line, and what is the slope of a vertical line?

Answer: Horizontal: $0$; vertical: undefined. Horizontal lines have no rise; vertical lines have no run.

Flashcard 32: Find the missing leg $a$ if a right triangle has hypotenuse $13$ and other leg $5$.

Answer: $12$. Use $a^2+5^2=13^2$, so $a^2=169-25=144$, thus $a=12$.

Flashcard 33: Find the area of a right triangle with legs $8$ and $11$.

Answer: $44$. Area = $\frac{1}{2} \times 8 \times 11 = 44$.

Flashcard 34: In a $30^\circ-60^\circ-90^\circ$ triangle, what is the long leg if the short leg is $x$?

Answer: $x\sqrt{3}$. Long leg (opposite 60°) = short leg × $\sqrt{3}$ in 30-60-90 triangles.

Flashcard 35: What is the hypotenuse $c$ if the legs are $9$ and $12$?

Answer: $15$. $9^2+12^2=81+144=225=15^2$ by Pythagorean theorem.

Flashcard 36: State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $x$.

Answer: $x:x:x\sqrt{2}$. Isosceles right triangle has equal legs and hypotenuse $\sqrt{2}$ times longer.

Flashcard 37: In a right triangle, what is the relationship between the two acute angles?

Answer: They are complementary: sum is $90^\circ$. The three angles sum to $180°$, with one being $90°$.

Flashcard 38: In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, find the hypotenuse if the short leg is $9$.

Answer: $18$. Hypotenuse is twice the short leg in 30-60-90 triangles.

Flashcard 39: Find the long leg in a $30^\circ$-$60^\circ$-$90^\circ$ triangle if the short leg is $4$.

Answer: $4\sqrt{3}$. Long leg = short leg × $\sqrt{3}$ in $30°$-$60°$-$90°$ triangle.

Flashcard 40: Find the short leg if a $30^\circ$-$60^\circ$-$90^\circ$ triangle has hypotenuse $18$.

Answer: $9$. Short leg = hypotenuse ÷ 2 in a 30-60-90 triangle.

Flashcard 41: What is the short leg if a $30^\circ$-$60^\circ$-$90^\circ$ triangle has hypotenuse $18$?

Answer: $9$. Short leg is half the hypotenuse in $30°$-$60°$-$90°$ triangle.

Flashcard 42: Which inequality classifies a triangle as acute using sides $a\le b<c$?

Answer: Acute if $a^2+b^2>c^2$. When leg squares exceed hypotenuse square, all angles are less than $90°$.

Flashcard 43: In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is the hypotenuse if each leg is $7$?

Answer: $7\sqrt{2}$. Apply the $45°$-$45°$-$90°$ ratio: hypotenuse = leg × $\sqrt{2}$.

Flashcard 44: State the definition of the altitude to the hypotenuse in a right triangle.

Answer: A segment from the right angle perpendicular to the hypotenuse. Creates a perpendicular from the vertex angle to the opposite side.

Flashcard 45: Identify whether side lengths $7,24,25$ form a right triangle.

Answer: Yes, because $7^2+24^2=25^2$. Check: $49+576=625$, which equals $25^2$.

Flashcard 46: What is the $30^\circ$-$60^\circ$-$90^\circ$ side ratio (short, long, hypotenuse)?

Answer: $1:\sqrt{3}:2$. Sides opposite $30°$, $60°$, and $90°$ angles follow this ratio.

Flashcard 47: What is the distance between points $(1,2)$ and $(5,5)$?

Answer: $5$. $d=\sqrt{(5-1)^2+(5-2)^2}=\sqrt{16+9}=\sqrt{25}=5$.

Flashcard 48: Find the missing leg $b$ if $c=13$ and $a=5$ in a right triangle.

Answer: $12$. Use $b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.

Flashcard 49: State the definition of a Pythagorean triple in terms of integers $a,b,c$ with $c$ largest.

Answer: Integers with $a^2+b^2=c^2$. Three positive integers satisfying the Pythagorean theorem.

Flashcard 50: State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$.

Answer: short $x$, long $x\sqrt{3}$, hypotenuse $2x$. Special triangle with sides in ratio $1:\sqrt{3}:2$.

Flashcard 51: State the side ratio for a $45^\circ-45^\circ-90^\circ$ right triangle in simplest radical form.

Answer: $1:1:\sqrt{2}$. Equal legs to hypotenuse ratio in an isosceles right triangle.

Flashcard 52: Find $c$ if a right triangle has legs $6$ and $8$.

Answer: $10$. Using $6^2+8^2=36+64=100$, so $c=\sqrt{100}=10$.

Flashcard 53: What is the shorter leg if a $30^\circ$-$60^\circ$-$90^\circ$ triangle has hypotenuse $2x$?

Answer: $x$. In a $30°$-$60°$-$90°$ triangle, shortest leg is half the hypotenuse.

Flashcard 54: What similarity relationship is created by drawing the altitude to the hypotenuse of a right triangle?

Answer: The $2$ smaller triangles are similar to the original and to each other. All three triangles share the same angle measures.

Flashcard 55: Find the hypotenuse $c$ if the legs are $9$ and $12$.

Answer: $15$. $c = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$.

Flashcard 56: State the side-length ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle (short:long:hypotenuse).

Answer: $1:\sqrt{3}:2$. Sides opposite $30°$, $60°$, and $90°$ angles respectively.

Flashcard 57: State the Pythagorean Theorem for a right triangle with legs $a,b$ and hypotenuse $c$.

Answer: $a^2+b^2=c^2$. Relates the squares of the legs to the square of the hypotenuse.

Flashcard 58: Identify the hypotenuse in a right triangle in terms of its position relative to the $90^\circ$ angle.

Answer: The side opposite the $90^\circ$ angle. The longest side in a right triangle, always opposite the right angle.

Flashcard 59: State the slope formula for a line through $(x_1,y_1)$ and $(x_2,y_2)$.

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run; change in $y$ over change in $x$.

Flashcard 60: State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$.

Answer: Sides $x,x\sqrt{3},2x$. Short leg : long leg : hypotenuse = $1 : \sqrt{3} : 2$.

Flashcard 61: Identify whether $7,24,25$ can be side lengths of a right triangle.

Answer: Yes, since $7^2+24^2=25^2$. Check if Pythagorean theorem holds: $49+576=625$ ✓.

Flashcard 62: What is the area formula for a right triangle with legs $a$ and $b$ as base and height?

Answer: $A=\frac{1}{2}ab$. Half the product of the two perpendicular sides.

Flashcard 63: State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle in simplest form.

Answer: $1:1:\sqrt{2}$. Isosceles right triangles have two equal legs and hypotenuse $\sqrt{2}$ times a leg.

Flashcard 64: Find $c$ for a right triangle with legs $6$ and $8$.

Answer: $10$. Apply $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.

Flashcard 65: What are the side lengths in the Pythagorean triple with legs $5$ and $12$?

Answer: $5,12,13$. The hypotenuse is $\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$.

Flashcard 66: Identify the hypotenuse in a right triangle in relation to the right angle.

Answer: The side opposite the $90^\circ$ angle. The longest side in a right triangle, always opposite the right angle.

Flashcard 67: State the Pythagorean triple obtained by scaling $3,4,5$ by a factor of $k$.

Answer: $3k,4k,5k$. Multiplying each side by $k$ preserves the right angle.

Flashcard 68: In a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $x$, what is the hypotenuse?

Answer: $x\sqrt{2}$. In isosceles right triangles, hypotenuse equals leg times $\sqrt{2}$.

Flashcard 69: State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle in simplest form.

Answer: $1:\sqrt{3}:2$. Short leg opposite $30°$, long leg opposite $60°$, hypotenuse opposite $90°$.

Flashcard 70: Find the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $9$.

Answer: $9\sqrt{2}$. In $45°$-$45°$-$90°$ triangle, hypotenuse equals leg times $\sqrt{2}$.

Flashcard 71: Find the hypotenuse of a $45^\circ-45^\circ-90^\circ$ triangle with leg $7$.

Answer: $7\sqrt{2}$. Multiply leg by $\sqrt{2}$ in a 45-45-90 triangle: $7\times\sqrt{2}$.

Flashcard 72: State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle in simplest form.

Answer: $1:\sqrt{3}:2$. Short leg : long leg : hypotenuse in this special triangle.

Flashcard 73: What is the hypotenuse if a $45^\circ$-$45^\circ$-$90^\circ$ triangle has leg $x$?

Answer: $x\sqrt{2}$. In a $45°$-$45°$-$90°$ triangle, hypotenuse equals leg times $\sqrt{2}$.

Flashcard 74: Find the missing leg $b$ if $c=13$ and $a=5$ in a right triangle.

Answer: $12$. Using $5^2+b^2=13^2$, so $b^2=169-25=144$, thus $b=12$.

Flashcard 75: State the side-length ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg length $x$.

Answer: Legs $x,x$; hypotenuse $x\sqrt{2}$. Isosceles right triangle has equal legs and hypotenuse $\sqrt{2}$ times longer.

Flashcard 76: State the side-length ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle.

Answer: $1:1:\sqrt{2}$. Legs are equal, hypotenuse is $\sqrt{2}$ times a leg.

Flashcard 77: Find the distance between $(1,2)$ and $(4,6)$.

Answer: $5$. $d=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=\sqrt{25}=5$.

Flashcard 78: Find the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg $x=9$.

Answer: $9\sqrt{2}$. In 45-45-90 triangle, hypotenuse = leg × $\sqrt{2}$.

Flashcard 79: Find the hypotenuse if a $45^\circ$-$45^\circ$-$90^\circ$ triangle has leg $7$.

Answer: $7\sqrt{2}$. In 45-45-90 triangle, hypotenuse = leg × $\sqrt{2}$.

Flashcard 80: What is the distance formula derived from the Pythagorean Theorem for points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses the Pythagorean theorem on the coordinate plane.

Flashcard 81: In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the short leg if the hypotenuse is $18$?

Answer: $9$. Short leg = hypotenuse ÷ 2 in a $30°$-$60°$-$90°$ triangle.

Flashcard 82: State the side ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle with leg length $x$.

Answer: Sides $x,x,x\sqrt{2}$. Isosceles right triangle has legs equal, hypotenuse $\sqrt{2}$ times longer.

Flashcard 83: In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, what is the hypotenuse in terms of short leg $x$?

Answer: $2x$. The hypotenuse is twice the side opposite $30°$.

Flashcard 84: Find the hypotenuse in a $30^\circ$-$60^\circ$-$90^\circ$ triangle if the short leg is $6$.

Answer: $12$. Hypotenuse = $2$ × short leg in $30°$-$60°$-$90°$ triangle.

Flashcard 85: What is the converse of the Pythagorean Theorem for side lengths $a,b,c$ with $c$ largest?

Answer: If $a^2+b^2=c^2$, the triangle is right. If sides satisfy the Pythagorean equation, then the angle is $90°$.

Flashcard 86: Find the distance between $(2,3)$ and $(8,11)$.

Answer: $10$. Distance formula: $\sqrt{(8-2)^2+(11-3)^2}=\sqrt{36+64}=\sqrt{100}$.

Flashcard 87: Find $c$ if a right triangle has legs $9$ and $12$.

Answer: $15$. Apply Pythagorean theorem: $c=\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$.

Flashcard 88: State the side ratios for a $30^\circ$-$60^\circ$-$90^\circ$ triangle (short, long, hypotenuse).

Answer: $x,\ x\sqrt{3},\ 2x$. Special triangle with sides in ratio $1:\sqrt{3}:2$.

Flashcard 89: State the Pythagorean identity relating $\sin(\theta)$ and $\cos(\theta)$.

Answer: $\sin^2(\theta)+\cos^2(\theta)=1$. Follows from dividing the Pythagorean theorem by the hypotenuse squared.

Flashcard 90: State the side ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$.

Answer: Sides $x,x\sqrt{3},2x$. Opposite $30°$, $60°$, $90°$ angles respectively.

Flashcard 91: State the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$ using a right triangle.

Answer: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Derived from the Pythagorean theorem applied to coordinate differences.

Flashcard 92: State the distance formula that comes from the Pythagorean Theorem for points $(x_1,y_1)$ and $(x_2,y_2)$.

Answer: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Pythagorean theorem applied to coordinate differences.

Flashcard 93: What is the missing leg $b$ if the hypotenuse is $13$ and the other leg is $5$?

Answer: $12$. $b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.

Flashcard 94: Find the area of a right triangle with legs $8$ and $15$.

Answer: $60$. Area = $\frac{1}{2} \times 8 \times 15 = 60$.

Flashcard 95: In a right triangle, if the legs are $6$ and $8$, what is the hypotenuse length?

Answer: $10$. Apply Pythagorean theorem: $6^2+8^2=36+64=100=10^2$.

Flashcard 96: State the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$ using a right triangle.

Answer: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses Pythagorean theorem on coordinate differences.

Flashcard 97: In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, what is the hypotenuse in terms of leg $x$?

Answer: $x\sqrt{2}$. In isosceles right triangles, hypotenuse equals leg times $\sqrt{2}$.

Flashcard 98: In a $30^\circ-60^\circ-90^\circ$ triangle, what is the short leg if the hypotenuse is $h$?

Answer: $\frac{h}{2}$. Short leg (opposite 30°) is half the hypotenuse in 30-60-90 triangles.

Flashcard 99: What is the converse of the Pythagorean Theorem for sides $a,b,c$ with $c$ largest?

Answer: If $a^2+b^2=c^2$, then the triangle is right. If the sum of squares of two sides equals the square of the third, it's a right triangle.

Flashcard 100: In a $30^\circ$-$60^\circ$-$90^\circ$ triangle with short leg $x$, what is the long leg?

Answer: $x\sqrt{3}$. Long leg equals short leg times $\sqrt{3}$ in a 30-60-90 triangle.