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Basic Geometry : Right Triangles

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Example Questions

Example Question #1181 : Plane Geometry

Triangles 3

Refer to the above diagram.

True or false: $\bigtriangleup AOB \sim \bigtriangleup COD$

Possible Answers:

False

True

Correct answer:

True

Explanation:

The distance from the origin to A (0, 12) is the absolute value of the -coordinate of , which is OA = 12 . Similarly, OB = 6 , OC = 16 , and OD = 8 . Also, since the axes intersect at right angles, $\angle AOB$ and $\angle COD$ are both right, and, consequently, congruent.

According to the Side-Angle-Side Similarity Theorem (SASS), if two sides of a triangle are in proportion to the corresponding sides of a second triangle, and their included angles are congruent, the triangles are similar.

We can test the proportion statement

$\frac{OA}{OC} = \frac{OB}{OD}$

by substituting:

$\frac{12}{16} = \frac{6}{8}$

Test the truth of this statement by comparing their cross products:

$12 \times 8 = 6 \times 16$

96 = 96

The cross-products are equal, making the proportion statement true, so two pairs of sides are in proportion. Also, their included angles $\angle AOB$ and $\angle COD$ are congruent. This sets up the conditions of SASS, so $\bigtriangleup AOB \sim \bigtriangleup COD$ .

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Example Question #11 : Right Triangles

Triangles

Refer to the above figure.

True, false, or inconclusive: $\bigtriangleup ADB \sim \bigtriangleup BDC$ .

Possible Answers:

False

True

Inconclusive

Correct answer:

True

Explanation:

$\overline{AD}$ is an altitude of $\bigtriangleup ABC$ , so it divides the triangle into two smaller triangles similar to each other - that is, if we match the shorter legs, the longer legs, and the hypotenuses, the similarity statement is

$\bigtriangleup ADB \sim \bigtriangleup BDC$ .

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

What is the length of the remaining side of the right triangle?

Possible Answers:

$14\ cm$

$6\ cm$

$10\ cm$

$12\ cm$

$7\ cm$

Correct answer:

$12\ cm$

Explanation:

Rearrange the Pythagorean Theorem to find the missing side. The Pythagorean Theorem is:

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

where is the hypotenuse and and are the sides.

$\sqrt{(15^{2}-9^{2})} = 12$

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Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

The hypotenuse of a right triangle is 26 in and one leg is 10 in. What is the sum of the two shortest sides?

Possible Answers:

$34\; in$

$28\; in$

$36\; in$

$30\; in$

$32\; in$

Correct answer:

$34\; in$

Explanation:

We use the Pythagorean Theorem so the problem to solve becomes $10^{2}+x^{2}=26^{2}$ where = unknown leg length

So $100 + x^{2}=676$ and $x=24\; in$

The sum of the two legs becomes $10+24=34\; in$

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Example Question #3 : How To Find The Length Of The Side Of A Right Triangle

$In\;\Delta KLM,\;what\;is\;the\;length\;of\;\overline{KL}?$

Possible Answers:

Correct answer:

Explanation:

$Use\;the\;Pythagorean\;Theorem: A^2+B^2=C^2$

$\fn_cm A^2+4^2=5^2$

A^2+16=25

A^2=25-16=9

$A=\sqrt9$

A=3

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Find the length of the unknown side, , of the right triangle below.

Triangle_a_9_12

Possible Answers:

$2\sqrt{5}$

$2\sqrt7$

$3\sqrt{10}$

$3\sqrt7$

$3\sqrt5$

Correct answer:

$3\sqrt7$

Explanation:

We need to use the Pythagorean Theorem, which says that a^2+b^2=c^2 .

Since we need to find the length of 'a', we can just solve for 'a'.

a^2=c^2-b^2

$a=\sqrt{c^2-b^2}$

In our case, c = 12 and b = 9.

Thus, $a=\sqrt{12^2-9^2}=\sqrt{144-81}=\sqrt{63}=\sqrt{9 \cdot 7}=\sqrt{9}\sqrt{7}=3\sqrt7$ .

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Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Find the missing leg of the right triangle when one of the legs is and the hypotenuse is .

Possible Answers:

Correct answer:

Explanation:

In order to find the missing side of a right triangle you must use one of two things:

1. Pythagorean Theorem

2. Trigonometry.

Since we only know what the side lengths are we must use the Pythagorean Theorem.

a^2 +b^2 =c^2

a=4, b=x, and c=5

(4)^2+x^2=(5)^2

16+x^2=25

x^2=9

x=3

So the missing side length is 3

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Given that the hypotenuse of a right triangle is and one side is , find the other side length.

Possible Answers:

144

Correct answer:

Explanation:

To find the length of the missing side, you can either use the pythagorean theorm or realize this is a case of a special right triangle with sides 5,12,13 .

Thus, the other side length is .

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Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Find the length of the missing side.

Possible Answers:

22.39

19.61

20.49

21.65

Correct answer:

20.49

Explanation:

Recall the Pythagorean Theorem for a right triangle:

a^2+b^2=c^2

Since the missing side corresponds to side , rewrite the Pythagorean Theorem and solve for .

b^2=c^2-a^2

$b=\sqrt{c^2-a^2}$

Now, plug in values of and into a calculator to find the length of side . Round to decimal places.

$b=\sqrt{22^2-8^2}=\sqrt{420}=20.49$

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Find the length of the missing side.

Possible Answers:

20.07

21.97

16.37

18.01

Correct answer:

20.07

Explanation:

Recall the Pythagorean Theorem for a right triangle:

a^2+b^2=c^2

Since the missing side corresponds to side , rewrite the Pythagorean Theorem and solve for .

b^2=c^2-a^2

$b=\sqrt{c^2-a^2}$

Now, plug in values of and into a calculator to find the length of side . Round to decimal places.

$b=\sqrt{22^2-p^2}=\sqrt{403}=20.07$

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Basic Geometry : Right Triangles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #1181 : Plane Geometry

Example Question #11 : Right Triangles

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Example Question #3 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

All Basic Geometry Resources

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