How to find the length of the side of a right triangle - PSAT Math

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Question

Which set of sides could make a right triangle?

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Answer

By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.

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Question

A right triangle with a base of 12 and hypotenuse of 15 is shown below. Find x.

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Answer

Using the Pythagorean Theorem, the height of the right triangle is found to be = √(〖15〗2 –〖12〗2) = 9, so x=9 – 5=4

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Question

A right triangle has sides of 36 and 39(hypotenuse). Find the length of the third side

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Answer

use the pythagorean theorem:

a2 + b2 = c2 ; a and b are sides, c is the hypotenuse

a2 + 1296 = 1521

a2 = 225

a = 15

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Question

A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:

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Answer

The Pythagorean Theorem gives us _a_2 + _b_2 = _c_2 for a right triangle, where c is the hypotenuse and a and b are the smaller sides. Here a is equal to 5 and c is equal to 14, so _b_2 = 142 – 52 = 171. Therefore b is equal to the square root of 171 or approximately 13.07.

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Question

Which of the following could NOT be the lengths of the sides of a right triangle?

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Answer

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.
All of the other answer choices observe the theorem _a_2 + _b_2 = _c_2

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Question

The area of a right traingle is 42. One of the legs has a length of 12. What is the length of the other leg?

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Answer

$Area= \frac{1}{2}\times base\times height$

$42=\frac{1}{2}\times base\times 12$

$42=6\times base$

$base=7$

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Question

$\left ( \bigtriangleup ABC \textup{ is a right triangle where } \angle B \textup{ is a right angle}\right)$

If $\angle A=30^{\circ}$ and $\overline{AB}=4$ , what is the length of $\overline{BC}$ ?

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Answer

AB is the leg adjacent to Angle A and BC is the leg opposite Angle A.

Since we have a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle, the opposites sides of those angles will be in the ratio $x-x\sqrt{3}-2x$ .

Here, we know the side opposite the sixty degree angle. Thus, we can set that value equal to $x\sqrt{3}$ .

$4=x\sqrt{3}$

$\frac{4}{\sqrt{3}}=\frac{x\sqrt{3}}{\sqrt{3}}$

$\frac{4}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=x$

$x=\frac{4\sqrt{3}}{3}$

which also means

$\overline{BC}=\frac{4\sqrt{3}}{3}$

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Question

Solve for x.

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Answer

Use the Pythagorean Theorem. Let a = 8 and c = 10 (because it is the hypotenuse)

$\small a^2+x^2=c^2$

$\small 8^2+x^2=10^2$

$\small 64+x^2=100$

$\small x^2=100-64=36$

$\small x=6$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

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Answer

The altitude perpendicular to the hypotenuse of a right triangle divides that triangle into two smaller triangles similar to each other and the large triangle. Therefore, the sides are in proportion. The hypotenuse of the triangle is equal to

$\sqrt{5^{2}+12^{2}} = \sqrt{25+144} = \sqrt{169} =13$

Therefore, we can set up, and solve for in, a proportion statement involving the shorter side and hypotenuse of the large triangle and the larger of the two smaller triangles:

$\frac{M}{5} = \frac{5}{13}$

$\frac{M}{5} \cdot 5 = \frac{5}{13} \cdot 5$

$M = \frac{25}{13} = 1\frac{12}{13}$

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Question

Bob the Helicopter is at 30,000 ft. above sea level, and as viewed on a map his airport is 40,000 ft. away. If Bob travels in a straight line to his airport at 250 feet per second, how many minutes will it take him to arrive?

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Answer

Draw a right triangle with a height of 30,000 ft. and a base of 40,000 ft. The hypotenuse, or distance travelled, is then 50,000ft using the Pythagorean Theorem. Then dividing distance by speed will give us time, which is 200 seconds, or 3 minutes and 20 seconds.

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Question

A right triangle has two sides, 9 and x, and a hypotenuse of 15. What is x?

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Answer

We can use the Pythagorean Theorem to solve for x.

92 + _x_2 = 152

81 + _x_2 = 225

_x_2 = 144

x = 12

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Question

Which set of sides could make a right triangle?

Tap to reveal answer

Answer

By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.

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Question

A right triangle with a base of 12 and hypotenuse of 15 is shown below. Find x.

Tap to reveal answer

Answer

Using the Pythagorean Theorem, the height of the right triangle is found to be = √(〖15〗2 –〖12〗2) = 9, so x=9 – 5=4

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Question

A right triangle has sides of 36 and 39(hypotenuse). Find the length of the third side

Tap to reveal answer

Answer

use the pythagorean theorem:

a2 + b2 = c2 ; a and b are sides, c is the hypotenuse

a2 + 1296 = 1521

a2 = 225

a = 15

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Question

A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:

Tap to reveal answer

Answer

The Pythagorean Theorem gives us _a_2 + _b_2 = _c_2 for a right triangle, where c is the hypotenuse and a and b are the smaller sides. Here a is equal to 5 and c is equal to 14, so _b_2 = 142 – 52 = 171. Therefore b is equal to the square root of 171 or approximately 13.07.

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Question

Which of the following could NOT be the lengths of the sides of a right triangle?

Tap to reveal answer

Answer

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.
All of the other answer choices observe the theorem _a_2 + _b_2 = _c_2

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Question

The area of a right traingle is 42. One of the legs has a length of 12. What is the length of the other leg?

Tap to reveal answer

Answer

$Area= \frac{1}{2}\times base\times height$

$42=\frac{1}{2}\times base\times 12$

$42=6\times base$

$base=7$

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Question

$\left ( \bigtriangleup ABC \textup{ is a right triangle where } \angle B \textup{ is a right angle}\right)$

If $\angle A=30^{\circ}$ and $\overline{AB}=4$ , what is the length of $\overline{BC}$ ?

Tap to reveal answer

Answer

AB is the leg adjacent to Angle A and BC is the leg opposite Angle A.

Since we have a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle, the opposites sides of those angles will be in the ratio $x-x\sqrt{3}-2x$ .

Here, we know the side opposite the sixty degree angle. Thus, we can set that value equal to $x\sqrt{3}$ .

$4=x\sqrt{3}$

$\frac{4}{\sqrt{3}}=\frac{x\sqrt{3}}{\sqrt{3}}$

$\frac{4}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=x$

$x=\frac{4\sqrt{3}}{3}$

which also means

$\overline{BC}=\frac{4\sqrt{3}}{3}$

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Question

Solve for x.

Tap to reveal answer

Answer

Use the Pythagorean Theorem. Let a = 8 and c = 10 (because it is the hypotenuse)

$\small a^2+x^2=c^2$

$\small 8^2+x^2=10^2$

$\small 64+x^2=100$

$\small x^2=100-64=36$

$\small x=6$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Tap to reveal answer

Answer

The altitude perpendicular to the hypotenuse of a right triangle divides that triangle into two smaller triangles similar to each other and the large triangle. Therefore, the sides are in proportion. The hypotenuse of the triangle is equal to

$\sqrt{5^{2}+12^{2}} = \sqrt{25+144} = \sqrt{169} =13$

Therefore, we can set up, and solve for in, a proportion statement involving the shorter side and hypotenuse of the large triangle and the larger of the two smaller triangles:

$\frac{M}{5} = \frac{5}{13}$

$\frac{M}{5} \cdot 5 = \frac{5}{13} \cdot 5$

$M = \frac{25}{13} = 1\frac{12}{13}$

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Knew It

How to find the length of the side of a right triangle - PSAT Math

Which set of sides could make a right triangle?

By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.

A right triangle with a base of 12 and hypotenuse of 15 is shown below. Find x.

Using the Pythagorean Theorem, the height of the right triangle is found to be = √(〖15〗2 –〖12〗2) = 9, so x=9 – 5=4

A right triangle has sides of 36 and 39(hypotenuse). Find the length of the third side

use the pythagorean theorem: a2 + b2 = c2 ; a and b are sides, c is the hypotenuse a2 + 1296 = 1521 a2 = 225 a = 15

A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:

The Pythagorean Theorem gives us _a_2 + _b_2 = _c_2 for a right triangle, where c is the hypotenuse and a and b are the smaller sides. Here a is equal to 5 and c is equal to 14, so _b_2 = 142 – 52 = 171. Therefore b is equal to the square root of 171 or approximately 13.07.

Which of the following could NOT be the lengths of the sides of a right triangle?

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100. All of the other answer choices observe the theorem _a_2 + _b_2 = _c_2

The area of a right traingle is 42. One of the legs has a length of 12. What is the length of the other leg?

If and , what is the length of ?

AB is the leg adjacent to Angle A and BC is the leg opposite Angle A. Since we have a triangle, the opposites sides of those angles will be in the ratio . Here, we know the side opposite the sixty degree angle. Thus, we can set that value equal to . which also means

Solve for x.

Use the Pythagorean Theorem. Let a = 8 and c = 10 (because it is the hypotenuse)

Note: Figure NOT drawn to scale. Refer to the above diagram. Evaluate .

Bob the Helicopter is at 30,000 ft. above sea level, and as viewed on a map his airport is 40,000 ft. away. If Bob travels in a straight line to his airport at 250 feet per second, how many minutes will it take him to arrive?

Draw a right triangle with a height of 30,000 ft. and a base of 40,000 ft. The hypotenuse, or distance travelled, is then 50,000ft using the Pythagorean Theorem. Then dividing distance by speed will give us time, which is 200 seconds, or 3 minutes and 20 seconds.

A right triangle has two sides, 9 and x, and a hypotenuse of 15. What is x?

We can use the Pythagorean Theorem to solve for x. 92 + _x_2 = 152 81 + _x_2 = 225 _x_2 = 144 x = 12

Which set of sides could make a right triangle?

By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.

A right triangle with a base of 12 and hypotenuse of 15 is shown below. Find x.

Using the Pythagorean Theorem, the height of the right triangle is found to be = √(〖15〗2 –〖12〗2) = 9, so x=9 – 5=4

A right triangle has sides of 36 and 39(hypotenuse). Find the length of the third side

use the pythagorean theorem: a2 + b2 = c2 ; a and b are sides, c is the hypotenuse a2 + 1296 = 1521 a2 = 225 a = 15

A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:

The Pythagorean Theorem gives us _a_2 + _b_2 = _c_2 for a right triangle, where c is the hypotenuse and a and b are the smaller sides. Here a is equal to 5 and c is equal to 14, so _b_2 = 142 – 52 = 171. Therefore b is equal to the square root of 171 or approximately 13.07.

Which of the following could NOT be the lengths of the sides of a right triangle?

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100. All of the other answer choices observe the theorem _a_2 + _b_2 = _c_2

The area of a right traingle is 42. One of the legs has a length of 12. What is the length of the other leg?

If and , what is the length of ?

AB is the leg adjacent to Angle A and BC is the leg opposite Angle A. Since we have a triangle, the opposites sides of those angles will be in the ratio . Here, we know the side opposite the sixty degree angle. Thus, we can set that value equal to . which also means

Solve for x.

Use the Pythagorean Theorem. Let a = 8 and c = 10 (because it is the hypotenuse)

Note: Figure NOT drawn to scale. Refer to the above diagram. Evaluate .

use the pythagorean theorem:

a2 + b2 = c2 ; a and b are sides, c is the hypotenuse

a2 + 1296 = 1521

a2 = 225

a = 15

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.
All of the other answer choices observe the theorem _a_2 + _b_2 = _c_2

$\left ( \bigtriangleup ABC \textup{ is a right triangle where } \angle B \textup{ is a right angle}\right)$

If $\angle A=30^{\circ}$ and $\overline{AB}=4$ , what is the length of $\overline{BC}$ ?

Use the Pythagorean Theorem. Let a = 8 and c = 10 (because it is the hypotenuse)

$\small a^2+x^2=c^2$

$\small 8^2+x^2=10^2$

$\small 64+x^2=100$

$\small x^2=100-64=36$

$\small x=6$

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

We can use the Pythagorean Theorem to solve for x.

92 + _x_2 = 152

81 + _x_2 = 225

_x_2 = 144

x = 12

use the pythagorean theorem:

a2 + b2 = c2 ; a and b are sides, c is the hypotenuse

a2 + 1296 = 1521

a2 = 225

a = 15

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.
All of the other answer choices observe the theorem _a_2 + _b_2 = _c_2

$\left ( \bigtriangleup ABC \textup{ is a right triangle where } \angle B \textup{ is a right angle}\right)$

If $\angle A=30^{\circ}$ and $\overline{AB}=4$ , what is the length of $\overline{BC}$ ?

Use the Pythagorean Theorem. Let a = 8 and c = 10 (because it is the hypotenuse)

$\small a^2+x^2=c^2$

$\small 8^2+x^2=10^2$

$\small 64+x^2=100$

$\small x^2=100-64=36$

$\small x=6$

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .