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

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Question

Given a right triangle with a leg length of 6 and a hypotenuse length of 10, find the length of the other leg, x.

Answer

Using Pythagorean Theorem, we can solve for the length of leg x:

_x_2 + 62 = 102

Now we solve for x:

_x_2 + 36 = 100

_x_2 = 100 – 36

_x_2 = 64

x = 8

Also note that this is proportionally a 3/4/5 right triangle, which is very common. Always look out for a side-to-hypoteneuse ratio of 3/5 or 4/5, or a side-to-side ratio of 3/4, in any right triangle, so that you may solve such triangles rapidly.

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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

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

In a right triangle a hypotenuse has a length of 8 and leg has a length of 7. What is the length of the third side to the nearest tenth?

Answer

Using the pythagorean theorem, 82=72+x2. Solving for x yields the square root of 15, which is 3.9

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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

Given a right triangle with a leg length of 2 and a hypotenuse length of √8, find the length of the other leg, x.

Answer

Using Pythagorean Theorem, we can solve for the length of leg x:

_x_2 + 22 = (√8)2 = 8

Now we solve for x:

_x_2 + 4 = 8

_x_2 = 8 – 4

_x_2 = 4

x = 2

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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

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

Points $\dpi{100} \small A$ , $\dpi{100} \small B$ , and $\dpi{100} \small C$ are collinear (they lie along the same line). $\angle ACD = 90^{\circ}$ , $\angle CAD=30^{\circ}$ , $\angle CBD=60^{\circ}$ , $\overline{AD}=4$

Find the length of segment $\overline{BD}$ .

Answer

The length of segment $\overline{BD}$ is $\frac{4\sqrt{3}}{3}$

Note that triangles $\dpi{100} \small ACD$ and $\dpi{100} \small BCD$ are both special, 30-60-90 right triangles. Looking specifically at triangle $\dpi{100} \small ACD$ , because we know that segment $\overline{AD}$ has a length of 4, we can determine that the length of segment $\overline{CD}$ is 2 using what we know about special right triangles. Then, looking at triangle $\dpi{100} \small BCD$ now, we can use the same rules to determine that segment $\overline{BD}$ has a length of $\frac{4}{\sqrt{3}}$

which simplifies to $\frac{4\sqrt{3}}{3}$ .

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Question

The legs of a right triangle are $8\ cm$ and $11\ cm$ . Rounded to the nearest whole number, what is the length of the hypotenuse?

Answer

Use the Pythagorean Theorem. The sum of both legs squared equals the hypotenuse squared.

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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 for (rounded to the nearest tenth). Figure not drawn to scale.

Answer

We will use the Pythagorean Theorem to solve for the missing side length.

$x=\sqrt{115}=10.7$

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Question

A square boxing ring has a perimeter of feet. When the two boxers are sitting in opposite corners between rounds, how far apart are they?

Answer

Since the perimeter of the ring is feet and the ring is a square, solve for the length of a single side of the ring by dividing by .

$\frac{80 \ feet \ perimeter}{4 \ sides} = 20 \ feet \ per \ side$

The distance between the two boxers in opposing corners is a straight line from any one corner to the other. That straight line forms the hypotenuse of a right triangle whose other two sides are each feet long (since they are each the sides of the square).

Solving for the length of the hypotenuse of this right triangle with the pythagorean theorem provides the distance between the two boxers when they are in opposite corners.

$(Side \ 1)^2 + (Side \ 2)^2 = (Hypotenuse)^2$

$\rightarrow (20)^2 + (20)^2 = (Hypotenuse)^2$

$\rightarrow 800 \ feet = (Hypotenuse)^2$

$\rightarrow \sqrt{800} \ feet = \sqrt{(Hypotenuse)^2}$

$\rightarrow 20\sqrt{2} \ feet = Hypotenuse$

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Question

Solve for .

(Figure not drawn to scale).

Answer

Use the Pythagorean theorem: $\small a^2+b^2=c^2$ .

We know the length of one side and the hypotenuse.

$\small q^2+24^2=26^2$

Now we can solve for the missing side.

$\small q^2+576=676$

$\small q^2=100$

$\small q=10$

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Question

Solve for $\small x$ .

Answer

Use the Pythagorean Theorem to solve for the missing side of the right triangle.

In this triangle, $\small a=x,\ b=12,\ c=15$ .

Now we can solve for $\small x$ .

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Question

Solve for $\small x$ .

Answer

This image depicts a 30-60-90 right triangle. The length of the side opposite the smallest angle is half the length of the hypotenuse.

$a^2+b^2=c^2\rightarrow (a)^2+(a\sqrt{3})^2=(2a)^2$

$a=\frac{220}{2}=110$

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