Pythagorean Theorem - GED Math

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Question

The two legs of a right triangle measure 30 and 40. What is its perimeter?

Answer

By the Pythagorean Theorem, if are the legs of a right triangle and is its hypotenuse,

$c^{2} = a^{2} + b^{2}$

Substitute and solve for :

$c^{2} = 30 ^{2} + 40^{2} = 900 + 1,600 = 2,500$

$c = \sqrt{2,500} = 50$

The perimeter of the triangle is

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Question

A right triangle has legs 30 and 40. Give its perimeter.

Answer

The hypotenuse of the right triangle can be calculated using the Pythagorean theorem:

$c = \sqrt{30^{2}+40^{2}} = \sqrt{900+1600} = \sqrt{2,500} = 50$

Add the three sides:

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Question

A right triangle has one leg measuring 14 inches; its hypotenuse is 50 inches. Give its perimeter.

Answer

The Pythagorean Theorem can be used to derive the length of the second leg:

$b = \sqrt{50^{2}-14^{2}} = \sqrt{2,500 - 196} = \sqrt{2,304} = 48$ inches

Add the three sides to get the perimeter.

inches.

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Question

A right triangle has one leg of length 42 inches; its hypotenuse has length 70 inches. What is the area of this triangle?

Answer

The Pythagorean Theorem can be used to derive the length of the second leg:

$b = \sqrt{70^{2}-42^{2}} = \sqrt{4,900-1,764} = \sqrt{3,136}= 56$ inches.

Use the area formula for a triangle, with the legs as the base and height:

$A = \frac{1}{2}bh = \frac{1}{2} \cdot 42 \cdot 70 = 1,176$ square inches.

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Question

Whicih of the following could be the lengths of the sides of a right triangle?

Answer

A triangle is right if and only if it satisfies the Pythagorean relationship

$a ^{2}+ b^{2}= c^{2}$

where is the measure of the longest side and are the other two sidelengths. We test each of the four sets of lengths, remembering to convert feet to inches by multiplying by 12.

7 inches, 2 feet, 30 inches:

2 feet is equal to 24 inches. The relationship to be tested is

$7^{2}+ 24 ^{2} = 30^{2}$

- False

10 inches, 1 foot, 14 inches:

1 foot is equal to 12 inches. The relationship to be tested is

$10^{2}+ 12^{2}= 14^{2}$

- False

15 inches, 3 feet, 40 inches:

3 feet is equal to 36 inches. The relationship to be tested is

$15^{2}+ 36^{2} = 40^{2}$

- False

2 feet, 32 inches, 40 inches:

2 feet is equal to 24 inches. The relationship to be tested is

$24 ^{2}+ 32^{2} = 40^{2}$

- True

The correct choice is 2 feet, 32 inches, 40 inches.

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Question

The area of a right triangle is 136 square inches; one of its legs measures 8 inches. What is the length of its hypotenuse? (If not exact, give the answer to the nearest tenth of an inch.)

Answer

The area of a triangle is calcuated using the formula

$A = \frac{1}{2} bh$

In a right triangle, the bases can be used for base and height, so solve for by substitution:

$136= \frac{1}{2} \cdot 8h$

$136 \div 4 =4h \div 4$

The legs measure 8 and 34 inches, respectively; use the Pythagorean Theorem to find the length of the hypotenuse:

$c=\sqrt{8^{2}+34^{2}} = \sqrt{64+1,156}= 34.9$ inches

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Question

An isosceles right triangle has hypotenuse 80 inches. Give its perimeter. (If not exact, round to the nearest tenth of an inch.)

Answer

Each leg of an isosceles right triangle has length that is the length of the hypotenuse divided by $\sqrt{2}$ . The hypotenuse has length 80, so each leg has length

$\frac{80}{\sqrt{2}} = \frac{80\cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}= \frac{80 \sqrt{2}}{2} = 40 \sqrt{2}$ .

The perimeter is the sum of the three sides:

$80 + 40 \sqrt{2}+ 40\sqrt{2}= 80 + 80\sqrt{2}$ inches.To the nearest tenth:

$80 + 80\sqrt{2} \approx 80 + 80 \times 1.414 \approx 80 + 113.1 = 193.1$ inches.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. . Give the perimeter of Quadrilateral .

Answer

The perimeter of Quadrilateral is the sum of the lengths of $\overline{AB}$ , $\overline{DE}$ , $\overline{BD}$ , and $\overline{EA}$ .

The first two lengths can be found by subtracting known lengths:

The last two segments are hypotenuses of right triangles, and their lengths can be calculated using the Pythagorean Theorem:

$\overline{BD }$ is the hypotenuse of a triangle with legs ; it measures

$BD = \sqrt{3^{2}+4^{2}} = \sqrt{9 + 16}= \sqrt{25} = 5$

$\overline{EA}$ is the hypotenuse of a triangle with legs ; it measures

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

Add the four sidelengths:

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

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

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

We can set up a proportion statement by comparing the large triangle to the smaller of the two in which it is divided. The sides compared are the hypotenuse and the longer side:

$\frac{X}{5} = \frac{12}{13}$

Solve for :

$\frac{X}{5} \cdot 5 = \frac{12}{13} \cdot 5$

$X = \frac{60}{13} = 4 \frac{8}{13}$

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Question

Refer to the above diagram. Which of the following expressions gives the length of $\overline{AC}$ ?

Answer

By the Pythagorean Theorem,

$AC = \sqrt{(AB)^{2} + (BC)^{2}}$

$AC = \sqrt{(x+10)^{2} + x^{2}}$

$AC = \sqrt{ x^{2}+ 2 \cdot x \cdot 10 +10^{2} + x^{2}}$

$AC = \sqrt{ x^{2}+ 2 \cdot x \cdot 10 +10^{2} + x^{2}}$

$AC = \sqrt{ x^{2}+20x +100+ x^{2}}$

$AC = \sqrt{ 2x^{2}+20x +100 }$

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Question

Note: figure NOT drawn to scale

The square in the above diagram has area 2,500. What is ?

Answer

The area of the square is 2,500; its sidelength is the square root of this:

$s = \sqrt{2,500} = 50$

This means that the right triangle in the diagram has legs and 40 and hypotenuse 50, so by the Pythagorean Theorem:

$x = \sqrt{50^{2} - 40^{2}} = \sqrt{2,500-1,600} = \sqrt{900} = 30$

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Question

The above figure shows Square with sidelength 2; is the midpoint of $\overline{SU}$ ; is the midpoint of $\overline{RS}$ . To the nearest tenth, what is the perimeter of Quadrilateral ?

Answer

The perimeter of Quadrilateral is the sum of the lengths of $\overline{QA}, \overline{AR}, \overline{RE}, \overline{EQ}$ .

$\overline{AR}$ is a side of the square and has length 2.

is the midpoint of $\overline{RS}$ , so $\overline{RE}$ is half a side of the square; its length is 1.

The other two sides are hypotenuses of right triangles and can be found using the Pythagorean Theorem.

and are the midpoints of $\overline{SU}$ and $\overline{RS}$ , respectively,both of which are segments with length 2; therefore,

.

$\overline{QE}$ is therefore the hypotenuse of a triangle with legs 1 and 1, and its length is

$QE = \sqrt{1^{2}+ 1^{2}} = \sqrt{1 + 1} = \sqrt{2}$ .

Similarly, $\overline{QA}$ is the hypotenuse of a triangle with legs 1 and 2, and its length is

$QE = \sqrt{1^{2}+ 2^{2}} = \sqrt{1 + 4} = \sqrt{5}$ .

Add these:

$P = 1 + 2 + \sqrt{2} + \sqrt{5} \approx 1 + 2 + 1.414 + 2.236 \approx 6.650$

This rounds to 6.7.

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Question

What is the area of a right triangle if the hypotenuse is 10, and one of the side lengths is 6?

Answer

To determine the other side length, we will need to use the Pythagorean Theorem.

Substitute the hypotenuse and the known side length as either or .

Subtract 36 from both sides and reduce.

Square root both sides and reduce.

The length and width of the triangle are now known.

Write the formula for the area of a triangle.

$A=\frac{1}{2}bh$

Substitute the dimensions.

$A=\frac{1}{2}(8)(6) = 24$

The answer is .

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Question

The hypotenuse of a right triangle is $43;\textup{cm}$ and one of its legs measures $22;\textup{cm}$ . What is the length of the triangle's other leg? Round to the nearest hundredth.

Answer

For this problem, you just need to remember your handy Pythagorean theorem. Remember that it is defined as:

where and are the legs of the triangle, and is the hypotenuse. Remember, however, that this only works for right triangles. Thus, based on your data, you know:

or

Subtracting 484 from each side of the equation, you get:

Using your calculator to calculate the square root, you get:

Rounding, this is , so the triangle's other leg measures $36.95;\text{cm}$ .

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Question

The hypotenuse of a right triangle is $1285;\text{in}$ and one of its leg measures $1028;\text{in}$ . What is the length of the triangle's other leg? Round to the nearest hundredth.

Answer

For this problem, you just need to remember your handy Pythagorean theorem. Remember that it is defined as:

where and are the legs of the triangle, and is the hypotenuse. Remember, however, that this only works for right triangles. Thus, based on your data, you know:

or

Subtracting 1056784 from each side of the equation, you get:

Using your calculator to calculate the square root, you get:

The length of the missing side of the triangle is $771;\text{in}$ .

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Question

If the hypotenuse of a right triangle is 5, and a side length is 2, what is the area?

Answer

To find the other side length, we will need to first use the Pythagorean Theorem.

Substitute the side and hypotenuse.

Solve for the missing side.

$a=\sqrt{21}$

Write the formula for the area of a triangle.

$A=\frac{BH}{2}$

Substitute the sides.

$A=\frac{2\sqrt{21}}{2} = \sqrt{21}$

The answer is: $\sqrt{21}$

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Question

If the hypotenuse of a right triangle if 7, and a side length is 5, what must be the length of the missing side?

Answer

Write the formula for the Pythagorean Theorem.

Substitute the values into the equation.

Subtract 25 from both sides.

Square root both sides.

$a = \sqrt{24} = \sqrt{4}\cdot \sqrt{6} = 2\sqrt{6}$

The answer is: $2\sqrt{6}$

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Question

Determine the hypotenuse of a right triangle if the side legs are respectively.

Answer

Write the Pythagorean Theorem to find the hypotenuse.

Substitute the dimensions.

Square root both sides.

$\sqrt{20}=\sqrt{c^2}$

$c= \sqrt{20} = \sqrt{4} \times \sqrt{5}$

The answer is: $2\sqrt5$

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Question

A car left City A and drove straight east for miles then it drove straight north for miles, where it stopped. In miles, what is the shortest distance between the car and City A?

Answer

Start by drawing out what the car did.

You'll notice that a right triangle will be created as shown by the figure above. Thus, the shortest distance between the car and City A is also the hypotenuse of the triangle. Use the Pythagorean Theorem to find the distance between the car and City A.

$\text{Distance}=\sqrt{60^2+80^2}=100$

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Question

You want to build a garden in the shape of a right triangle. If the two arms will be 6ft and 8ft, what does the length of the hypotenuse need to be?

Answer

You want to build a garden in the shape of a right triangle. If the two arms will be 6ft and 8ft, what does the length of the hypotenuse need to be?

To find the length of a hypotenuse of a right triangle, simply use the Pythagorean Theorem.

Where a and b are the arm lengths, and c is the hypotenuse.

Plug in our knowns and solve.

$c=\sqrt{100ft^2}=10ft$

Note that we could also have found c by identifying a Pythagorean Triple:

3x-4x-5x

3(2)-4(2)-5(2)

6-8-10

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