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Basic Geometry : Triangles

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Example Questions

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

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 #231 : Triangles

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 #38 : Right Triangles

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 #39 : Right Triangles

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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Example Question #23 : How To Find The Length Of The Side Of A Right Triangle

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle B$ is an acute angle; $\angle E$ is a right angle.

$\overline{AB} \cong \overline{DE}$

$\overline{BC} \cong \overline{EF}$

Which is a true statement?

Possible Answers:

AC < DF

AC = DF

AC > DF

Correct answer:

AC < DF

Explanation:

$\overline{AB} \cong \overline{DE}$ and $\overline{BC} \cong \overline{EF}$ . However, the included angle of $\overline{AB}$ and $\overline{BC}$ , $\angle B$ , is acute, so its measure is less than that of $\angle E$ , which is right. This sets up the conditions of the SAS Inequality Theorem (or Hinge Theorem); the side of lesser length is opposite the angle of lesser measure. Consequently, AC < DF .

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Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Trig_id

If o=10ft and a=17ft , how long is side ?

Possible Answers:

18ft

17.9ft

Not enough information to solve

19.7 ft

19.3ft

Correct answer:

19.7 ft

Explanation:

This problem is solved using the Pythagorean theorem $(a^{2}+b^{2}=c^{2})$ . In this formula and are the legs of the right triangle while is the hypotenuse.

Using the labels of our triangle we have:

$o^{2}+a^{2}=h^{2}$

$\dpi{100} h^{2}=(10ft)^{2} +(17ft)^{2}$

$h^{2}=100ft^{2}+289ft^{2}$

$h=\sqrt{389ft^{2}}$

$\rightarrow 19.7 ft$

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Example Question #2 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

If one of the short sides of a 45-45-90 triangle equals 5, how long is the hypotenuse?

Possible Answers:

5√2

√10

√15

Correct answer:

5√2

Explanation:

Using the Pythagorean theorem, x² + y² = h². And since it is a 45-45-90 triangle the two short sides are equal. Therefore 5² + 5² = h². Multiplied out 25 + 25 = h².

Therefore h² = 50, so h = √50 = √2 * √25 or 5√2.

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Example Question #391 : Plane Geometry

The height of a right circular cylinder is 10 inches and the diameter of its base is 6 inches. What is the distance from a point on the edge of the base to the center of the entire cylinder?

Possible Answers:

4π/5

3π/4

None of the other answers

√(34)

√(43)/2

Correct answer:

√(34)

Explanation:

The best thing to do here is to draw diagram and draw the appropiate triangle for what is being asked. It does not matter where you place your point on the base because any point will produce the same result. We know that the center of the base of the cylinder is 3 inches away from the base (6/2). We also know that the center of the cylinder is 5 inches from the base of the cylinder (10/2). So we have a right triangle with a height of 5 inches and a base of 3 inches. So using the Pythagorean Theorem 3²+ 5²= c². 34 = c², c = √(34).

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Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

A right triangle with sides A, B, C and respective angles a, b, c has the following measurements.

Side A = 3in. Side B = 4in. What is the length of side C?

Possible Answers:

Correct answer:

Explanation:

The correct answer is 5. The pythagorean theorem states that a²+ b²= c². So in this case 3²+ 4²= C². So C²= 25 and C = 5. This is also an example of the common 3-4-5 triangle.

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Example Question #4 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

The lengths of the three sides of a right triangle form a set of consecutive even integers when arranged from least to greatest. If the second largest side has a length of x, then which of the following equations could be used to solve for x?

Possible Answers:

(x – 2)² + x² = (x + 2)²

(x + 2)² + (x – 2)² = x²

(x – 1)² + x² = (x + 1)²

(x – 2) + x = (x + 2)

x ² + (x + 2)² = (x + 4)²

Correct answer:

(x – 2)² + x² = (x + 2)²

Explanation:

We are told that the lengths form a series of consecutive even integers. Because even integers are two units apart, the side lengths must differ by two. In other words, the largest side length is two greater than the second largest, and the second largest length is two greater than the smallest length.

The second largest length is equal to x. The second largest length must thus be two less than the largest length. We could represent the largest length as x + 2.

Similarly, the second largest length is two larger than the smallest length, which we could thus represent as x – 2.

To summarize, the lengths of the triangle (in terms of x) are x – 2, x, and x + 2.

In order to solve for x, we can make use of the fact that the triangle is a right triangle. If we apply the Pythagorean Theorem, we can set up an equation that could be used to solve for x. The Pythagorean Theorem states that if a and b are the lengths of the legs of the triangle, and c is the length of the hypotenuse, then the following is true:

a² + b² = c²

In this particular case, the two legs of our triangle are x – 2 and x, since the legs are the two smallest sides; therefore, we can say that a = x – 2, and b = x. Lastly, we can say c = x + 2, since x + 2 is the length of the hypotenuse. Subsituting these values for a, b, and c into the Pythagorean Theorem yields the following:

(x – 2)² + x² = (x + 2)²

The answer is (x – 2)² + x² = (x + 2)².

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Basic Geometry : Triangles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

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

Example Question #231 : Triangles

Example Question #38 : Right Triangles

Example Question #39 : Right Triangles

Example Question #23 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #2 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #391 : Plane Geometry

Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #4 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

All Basic Geometry Resources

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