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Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

Example Question #1211 : Basic Geometry

Tri 8

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

Possible Answers:

5.5

$\sqrt{42}$

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 #221 : Triangles

Tri 9

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

Possible Answers:

15.912

8.476

5.196

11.273

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 #35 : Right Triangles

Tri 11

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

Possible Answers:

32.985

71.76

40.95

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 #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:

True

False

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 : Plane 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:

False

True

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 #14 : Triangles

Trig_id

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

Possible Answers:

19.7 ft

19.3ft

17.9ft

18ft

Not enough information to solve

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 #1 : 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:

√10

√15

5√2

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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Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #1211 : Basic Geometry

Example Question #221 : Triangles

Example Question #35 : Right Triangles

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

Example Question #1211 : Plane Geometry

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 #14 : Triangles

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

All Basic Geometry Resources

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