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

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Question

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

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

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?

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?

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

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

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?

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

Which set of sides could make a right triangle?

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.

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

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?

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 $\overline{BC}$ ?

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.

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

Solve each problem and decide which is the best of the choices given.

If $sin x =\frac{12}{13}$ , what is ?

Answer

This is a triangle. We can find the value of the other leg by using the Pythagorean Theorem.

Remembering that

$sin=\frac{opposite}{hypotenuse}$ .

Thus,

.

If $sin(x)=\frac{12}{13}$ , you know the adjacent side is .

Thus, making

$tan(x)=\frac{opposite}{adjacent}=\frac{12}{5}$ because tangent is opposite/adjacent.

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Question

Given $\bigtriangleup ABC$ with $m \angle A > m \angle B$ and $m \angle C = 90^{\circ }$ .

Which of the following could be the correct ordering of the lengths of the sides of the triangle?

I)

II)

III)

Answer

Given two angles of unequal measure in a triangle, the side opposite the greater angle is longer than the side opposite the other angle. Since $m \angle A > m \angle B$ , it follows that . Also, in a right triangle, the hypotenuse must be the longest side; in $\bigtriangleup ABC$ , since $\angle C$ is the right side, this hypotenuse is $\overline{AB}$ . It follows that , and that (I) is the only statement that can possibly be true.

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Question

A single-sided ladder is leaning against a wall. The angle between the end of the ladder that is on the ground and the ground itself is represented by $x^{\circ}$ . The ladder is sliding down the wall at a rate of 6 feet per second. If

$\sin x^{\circ}=\frac{3}{5}$

how many seconds does it take for the ladder to fall all the way to the ground? (The wall is a right angle to the ground.)

Answer

The ladder leaning against the wall forms a right triangle. The hypotenuse of the triangle is 5 ft., the length of the ladder.

Because sin x= opposite/hypotenuse, sine of the angle is equal to the length of the side opposite the angle divided by the hypotenuse. In this case, the length of the side opposite the angle is h, the height of the end of the ladder that is touching the wall. Thus,

$\sin x^{\circ} = \frac{h}{5}$

Because we are told that

$\sin x^{\circ}=\frac{3}{5}$

we know that h=3. Therefore, 3 feet is the height of the ladder. If the ladder is falling at a rate of 6 feet per second, we can find the number of seconds it will take the ladder to hit the ground with the equation

where h represents the height the ladder is falling from, and s represents the number of seconds it takes the ladder to fall. We can now solve for s:

$s=\frac{h}{6}$

$s=\frac{3}{6}=\frac{1}{2}=0.5$

It takes the ladder 0.5 seconds to fall to the ground.

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Question

Refer to the provided figure. Give the length of $\overline{BC}$ .

Answer

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is $90^{\circ }$ , so

$m \angle A+ m \angle C = 90^{\circ }$

Substituting $45^{\circ }$ for $m \angle C$ :

$m \angle A+ 45^{\circ } = 90^{\circ }$

$m \angle A= 45^{\circ }$

This makes $\bigtriangleup ABC$ a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, the length of leg $\overline{BC}$ is equal to that of hypotenuse $\overline{AC}$ , the length of which is 20. Therefore,

$BC = \frac{AC}{\sqrt{2}} = \frac{20}{\sqrt{2}}$

Rationalize the denominator by multiplying both halves of the fraction by $\sqrt{2}$ :

$BC = \frac{20 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{20 \sqrt{2}}{2} = 10 \sqrt{2}$

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Question

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

Answer

use the pythagorean theorem:

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

a2 + 1296 = 1521

a2 = 225

a = 15

Compare your answer with the correct one above

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?

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.

Compare your answer with the correct one above

Question

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

Answer

We can use the Pythagorean Theorem to solve for x.

92 + _x_2 = 152

81 + _x_2 = 225

_x_2 = 144

x = 12

Compare your answer with the correct one above

Question

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

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.

Compare your answer with the correct one above

Question

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

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

Compare your answer with the correct one above

Question

Which set of sides could make a right triangle?

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.

Compare your answer with the correct one above

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