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Basic Geometry : Right Triangles

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Example Questions

Example Question #31 : Right Triangles

Tri 3

Given the right triangle above, find the value of .

Possible Answers:

Correct answer:

Explanation:

To find the length of the side x, we must use the Pythagorean Theorem

a^2+b^2=c^2 .

However, this time since we are given the value of the hypotenuse, we will solve for side b rather than c.

So, when we plug the given values into the formula, the equation looks like

4^2+b^2=5^2

which can be simplified to

16+b^2=25 .

Next, solve for b and we get a final answer of

$b=\sqrt{9}=3$ .

This particular example is a Pythagorean triple, or a right triangle with 3 whole number values, so it is a good one to remember.

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Example Question #1201 : Basic Geometry

Tri 5

Given the right triangle above, find the length of the missing side.

Possible Answers:

Correct answer:

Explanation:

To find the length of the side x, we must use the Pythagorean Theorem

a^2+b^2=c^2 .

However, this time since we are given the value of the hypotenuse, we will solve for side b rather than c.

So, when we plug the given values into the formula, the equation looks like

9^2+b^2=41^2

which can be simplified to

81+b^2=1681 .

Next, solve for b and we get a final answer of

$b=\sqrt{1600}=40$ .

This particular example is a Pythagorean triple, or a right triangle with 3 whole number values, so it is a good one to remember.

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Example Question #1201 : Basic Geometry

Tri 6

Given the right triangle above, find the length of the missing side.

Possible Answers:

Correct answer:

Explanation:

To find the length of the side x, we must use the Pythagorean Theorem

a^2+b^2=c^2 .

However, this time since we are given the value of the hypotenuse, we will solve for side b rather than c.

So, when we plug the given values into the formula, the equation looks like

20^2+b^2=29^2

which can be simplified to

400+b^2=841 .

Next, solve for b and we get a final answer of

$b=\sqrt{441}=21$ .

This particular example is a Pythagorean triple, or a right triangle with 3 whole number values, so it is a good one to remember.

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Example Question #1211 : Plane Geometry

Tri 8

Given the above right triangle, find the length of the missing side.

Possible Answers:

$\sqrt{42}$

5.5

Correct answer:

Explanation:

To find the length of the side x, we must use the Pythagorean Theorem

a^2+b^2=c^2 .

However, this time since we are given the value of the hypotenuse, we will solve for side b rather than c.

So, when we plug the given values into the formula, the equation looks like

$4^2+b^2=(\sqrt{41})^2$

which can be simplified to

16+b^2=41 .

Next, solve for b and we get a final answer of

$b=\sqrt{25}=5$ .

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Example Question #1211 : Basic Geometry

Tri 9

Find the length of the missing side of the right triangle.

Possible Answers:

15.912

8.476

11.273

5.196

17.234

Correct answer:

17.234

Explanation:

To find the length of the side x, we must use the Pythagorean Theorem

a^2+b^2=c^2 .

However, this time since we are given the value of the hypotenuse, we will solve for side b rather than c.

So, when we plug the given values into the formula, the equation looks like

8^2+b^2=19^2

which can be simplified to

64+b^2=361 .

Next, solve for b and we get a final answer of

$b=\sqrt{297}\approx17.234$ .

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Example Question #1211 : Plane Geometry

Tri 11

Find the length of the missing side of the right triangle.

Possible Answers:

40.95

71.76

32.985

Correct answer:

32.985

Explanation:

To find the length of the side x, we must use the Pythagorean Theorem

a^2+b^2=c^2 .

However, this time since we are given the value of the hypotenuse, we will solve for side b rather than c.

So, when we plug the given values into the formula, the equation looks like

64^2+b^2=72^2

which can be simplified to

4096+b^2=5184 .

Next, solve for b and we get a final answer of

$b=\sqrt{1088}\approx32.985$ .

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Example Question #1212 : Basic Geometry

The three sides of a triangle have lengths $\frac{1}{3}$ , $\frac{1}{4}$ , and $\frac{1}{5}$ .

True or false: the triangle is a right triangle.

Possible Answers:

False

True

Correct answer:

False

Explanation:

We can rewrite each of these fractional lengths in terms of their least common denominator, which is LCM (3, 4, 5) = 60 , as follows:

$\frac{1}{3} = \frac{1 \times 20 }{3 \times 20 } = \frac{20}{60 }$

$\frac{1}{4} = \frac{1 \times 15 }{4 \times 15} = \frac{15}{60 }$

$\frac{1}{5} = \frac{1 \times 12}{5 \times 12} = \frac{12}{60 }$

By the Pythagorean Theorem and its converse, a triangle is right if and only if

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

where is the length of the longest side and and are the lengths of the other two sides.

Therefore, we can set $a=\frac{12}{60 }, b= \frac{15}{60 }, c = \frac{20}{60 }$ , and test the truth of the statement:

$\left (\frac{12}{60 } \right ) ^{2}+ \left (\frac{15}{60 } \right )^{2}= \left ( \frac{20}{60 } \right )^{2}$

$\frac{144}{3,600} + \frac{225}{3,600} = \frac{400}{3,600}$

$\frac{369}{3,600} = \frac{400}{3,600}$

The statement is false, so the Pythagorean relationship does not hold. The triangle is not right.

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Example Question #1211 : Basic Geometry

The three sides of a triangle have lengths 0.8, 1.2, and 1.5.

True or false: the triangle is a right triangle.

Possible Answers:

True

False

Correct answer:

False

Explanation:

By the Pythagorean Theorem and its converse, a triangle is right if and only if

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

where is the length of the longest side and and are the lengths of the other two sides.

Therefore, set a = 0.8, b= 1.2, c = 1.5 and test the statement for truth or falsity:

$0.8^{2}+1.2^{2}= 1.5^{2}$

0.64+ 1.44 = 2.25

2.08 = 2.25

The statement is false, so the Pythagorean relationship does not hold. The triangle is not right.

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Example Question #21 : How To Find The Length Of The Side Of A Right Triangle

The three sides of a triangle have lengths $\sqrt{11}$ , $\sqrt{15}$ , and $\sqrt{26}$ .

True or false: the triangle is a right triangle.

Possible Answers:

False

True

Correct answer:

True

Explanation:

By the Pythagorean Theorem and its converse, a triangle is right if and only if

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

where is the length of the longest side and and are the lengths of the other two sides.

Therefore, we can set $a= \sqrt{11}, b= \sqrt{15}, c = \sqrt{26}$ , and test the truth of the statement:

$\left ( \sqrt{11} \right )^{2}+ \left ( \sqrt{15} \right )^{2}= \left ( \sqrt{26} \right )^{2}$

11+ 15 = 26

26=26

The statement is true, so the Pythagorean relationship holds. The triangle is right.

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Example Question #1211 : Plane Geometry

Shape area right triangle

In the right triangle shown here, a= 7 and c=25 . What is the length of the base ?

Possible Answers:

Correct answer:

Explanation:

Given the lengths of two sides of a right triangle, it is always possible to calculate the length of the third side using the Pythagorean Theorem:

a^2+b^2=c^2

Here, the given side lengths are a=7 and c=25 . Solving for yields:

(7)^2+b^2=(25)^2

49+b^2=625

b^2=625-49=576

$\Rightarrow b = \sqrt {576} = 24$ .

Hence, the length of the base of the given right triangle is units.

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Basic Geometry : Right Triangles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #31 : Right Triangles

Example Question #1201 : Basic Geometry

Example Question #1201 : Basic Geometry

Example Question #1211 : Plane Geometry

Example Question #1211 : Basic Geometry

Example Question #1211 : Plane Geometry

Example Question #1212 : Basic Geometry

Example Question #1211 : Basic Geometry

Example Question #21 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1211 : Plane Geometry

All Basic Geometry Resources

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