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GED Math : Pythagorean Theorem

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Example Question #1 : Pythagorean Theorem

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

Possible Answers:

140

130

125

120

135

Correct answer:

120

Explanation:

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

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

Substitute a = 30, b= 40 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

P = 30 + 40 + 50 = 120

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Example Question #2 : Pythagorean Theorem

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

Possible Answers:

120

1,200

600

Correct answer:

120

Explanation:

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:

P = 30 + 40 + 50 = 120

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Example Question #3 : Pythagorean Theorem

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

Possible Answers:

$64\textrm{ in}$

$114\textrm{ in}$

$112 \textrm{ in}$

$78 \textrm{ in}$

Correct answer:

$112 \textrm{ in}$

Explanation:

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.

P = 14 + 48 + 50 = 112 inches.

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Example Question #4 : Pythagorean Theorem

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

Possible Answers:

10 inches, 1 foot, 14 inches

15 inches, 3 feet, 40 inches

2 feet, 32 inches, 40 inches

7 inches, 2 feet, 30 inches

Correct answer:

2 feet, 32 inches, 40 inches

Explanation:

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 a,b 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

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

49 + 576 = 900

625 = 900 - False

10 inches, 1 foot, 14 inches:

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

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

100 + 144 = 196

244 = 196 - False

15 inches, 3 feet, 40 inches:

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

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

225 + 1,296 = 1,600

1,521 = 1,600 - False

2 feet, 32 inches, 40 inches:

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

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

576+ 1,024= 1,600

1,600 =1,600 - True

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

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Example Question #5 : Pythagorean Theorem

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

Possible Answers:

$153.1 \textrm{ in}$

$193.1 \textrm{ in}$

$96.6\textrm{ in}$

$136.6\textrm{ in}$

Correct answer:

$193.1 \textrm{ in}$

Explanation:

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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Example Question #6 : Pythagorean Theorem

Triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. AC = 5, BC = 3, CD = 4, CE = 12 . Give the perimeter of Quadrilateral ABDE .

Possible Answers:

Correct answer:

Explanation:

The perimeter of Quadrilateral ABDE 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:

AB = AC - BC = 5 - 3 = 2

DE = CE - DE = 12 - 4 = 8

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 BC = 3, CD = 4 ; it measures

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

$\overline{EA}$ is the hypotenuse of a triangle with legs AC = 5, CE = 12 ; it measures

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

Add the four sidelengths:

P = AB + BD + DE + EA = 2+ 5 + 8 + 13 = 28

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Example Question #7 : Pythagorean Theorem

Triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Possible Answers:

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

$X = 4\frac{7}{13}$

$X = 4\frac{2}{13}$

$X = 3\frac{11}{13}$

Correct answer:

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

Explanation:

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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Example Question #8 : Pythagorean Theorem

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

Possible Answers:

$1,470 \textrm{ in}^{2}$

$2,352\textrm{ in}^{2}$

$1,176 \textrm{ in}^{2}$

$2,940\textrm{ in}^{2}$

Correct answer:

$1,176 \textrm{ in}^{2}$

Explanation:

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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Example Question #9 : Pythagorean Theorem

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.)

Possible Answers:

$33.1\textrm{ in}$

$34\textrm{ in}$

$34.9\textrm{ in}$

$36\textrm{ in}$

Correct answer:

$34.9\textrm{ in}$

Explanation:

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=4h

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

34 = h

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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Example Question #10 : Pythagorean Theorem

Triangle

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

Possible Answers:

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

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

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

$AC = x^{2}+10x$

Correct answer:

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

Explanation:

By the Pythagorean Theorem,

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

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

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

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

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

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GED Math : Pythagorean Theorem

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #1 : Pythagorean Theorem

Example Question #2 : Pythagorean Theorem

Example Question #3 : Pythagorean Theorem

Example Question #4 : Pythagorean Theorem

Example Question #5 : Pythagorean Theorem

Example Question #6 : Pythagorean Theorem

Example Question #7 : Pythagorean Theorem

Example Question #8 : Pythagorean Theorem

Example Question #9 : Pythagorean Theorem

Example Question #10 : Pythagorean Theorem

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