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High School Math : Plane Geometry

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Example Questions

Example Question #18 : How To Find The Length Of The Side Of A Right Triangle

Solve for $\small x$ .

Possible Answers:

Correct answer:

Explanation:

Use the Pythagorean Theorem to solve for the missing side of the right triangle.

a^2+b^2=c^2

In this triangle, $\small a=x,\ b=12,\ c=15$ .

x^2+12^2=15^2

Now we can solve for $\small x$ .

x^2+144=225

x^2=81

x=9

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Example Question #14 : How To Find The Length Of The Side Of A Right Triangle

Solve for $\small x$ .

Possible Answers:

110

$55\sqrt2$

$110\sqrt3$

$110\sqrt2$

Correct answer:

110

Explanation:

This image depicts a 30-60-90 right triangle. The length of the side opposite the smallest angle is half the length of the hypotenuse.

$a^2+b^2=c^2\rightarrow (a)^2+(a\sqrt{3})^2=(2a)^2$

2a=220

$a=\frac{220}{2}=110$

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Example Question #19 : How To Find The Length Of The Side Of A Right Triangle

Given a right triangle, solve for the missing leg if one leg is 12 and the hypotenuse is 13.

Possible Answers:

17.69

Correct answer:

Explanation:

Since the traingle is a right traingle, we can use the Pythagorean Theorem to solve for the missing leg:

a^2 + b^2 = c^2

a=12 and the hypotenuse c=13

(12)^2 + b^2 = (13)^2

b^2 = (13)^2- (12)^2

b^2 = 169- 144

b^2 = 25

b = 5

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

Points $\dpi{100} \small A$ , $\dpi{100} \small B$ , and $\dpi{100} \small C$ are collinear (they lie along the same line). $\angle ACD = 90^{\circ}$ , $\angle CAD=30^{\circ}$ , $\angle CBD=60^{\circ}$ , $\overline{AD}=4$

Find the length of segment $\overline{BD}$ .

Possible Answers:

$\frac{2\sqrt{3}}{3}$

$2\sqrt{3}$

$2$

$\frac{\sqrt{3}}{2}$

$\frac{4\sqrt{3}}{3}$

Correct answer:

$\frac{4\sqrt{3}}{3}$

Explanation:

The length of segment $\overline{BD}$ is $\frac{4\sqrt{3}}{3}$

Note that triangles $\dpi{100} \small ACD$ and $\dpi{100} \small BCD$ are both special, 30-60-90 right triangles. Looking specifically at triangle $\dpi{100} \small ACD$ , because we know that segment $\overline{AD}$ has a length of 4, we can determine that the length of segment $\overline{CD}$ is 2 using what we know about special right triangles. Then, looking at triangle $\dpi{100} \small BCD$ now, we can use the same rules to determine that segment $\overline{BD}$ has a length of $\frac{4}{\sqrt{3}}$

which simplifies to $\frac{4\sqrt{3}}{3}$ .

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

$\textup{What is the length of the hypotenuse of a right triangle with legs of lengths 10 and 15?}$

Possible Answers:

$6\sqrt{13}$

$13\sqrt{5}$

$5\sqrt{13}$

Correct answer:

$5\sqrt{13}$

Explanation:

$\textup{Pythagorean Theorum: }a^{2}+b^{2}= c^{2}\;\;\;\;\;\;10^{2}+15^{2}=c^{2}$

$100+225=c^{2}\;\;\;\;325=c^{2}\;\;\;\;\;\sqrt{325}=c\;\;\;\;\sqrt{25}\times\sqrt{13}=c$

$5\sqrt{13}=c$

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

Rt_triangle_letters

If angle $c=90^\circ$ , Y=3 and Z=4 , what is the value of ?

Possible Answers:

12.5

Correct answer:

Explanation:

Once we see that $c=90^\circ$ , we know that we're working with a right triangle and that will be the hypotenuse.

At this point we can use the Pythaogrean theorem ( a^2+b^2=c^2 ) or, in this case: Y^2+Z^2=X^2 .

Plug in our given values to solve:

Y^2+Z^2=X^2

3^2+4^2=X^2

9+16=X^2

25=X^2

$\sqrt{25}=\sqrt{X^2}$

5=X

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

Rt_triangle_letters

If angle $c=90^\circ$ , X = 13 and Y = 12 , what is the value of ?

Possible Answers:

17.69

144

169

Correct answer:

Explanation:

Once we see that $c=90^\circ$ , we know we're working with a right triangle and that will be the hypotenuse.

At this point we can use the Pythaogrean theorem ( a^2+b^2=c^2 ) or, in this case: Y^2+Z^2=X^2 .

Plug in our given values to solve:

Y^2+Z^2=X^2

12^2+Z^2=13^2

144+Z^2=169

Subtract 144 from both sides:

Z^2=25

$\sqrt{Z^2}=\sqrt{25}$

Z=5

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

A right triangle has legs of and . What is the hypotenuse?

Possible Answers:

$5\sqrt{5}$

$\sqrt{5}$

$5\sqrt{2}$

Correct answer:

$5\sqrt{2}$

Explanation:

To solve this problem, use the Pythagorean theorem: the sum of the square of the legs equals the square of the hyoptenuse or, mathematically, a^2+b^2=c^2 .

Plug in our given values.

a^2+b^2=c^2

5^2+5^2=c^2

25+25=c^2

50=c^2

$\sqrt{50}=\sqrt{c^2}$

is not a perfect square, but let's see if we can find a factor that is a perfect square.

$\sqrt{25*2}=\sqrt{c^2}$

IS a perfect square, so we can simplify!

$\sqrt{25}*\sqrt{2}=\sqrt{c^2}$

$5*\sqrt2=c$

$5\sqrt{2}=c$

This is going to be true of all isoceles right triangles: the pattern will always be $x:x:x\sqrt{2}$ .

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

A right triangle has legs of and . What is the hypotenuse?

Possible Answers:

$2\sqrt{2}$

$4\sqrt{2}$

Correct answer:

$4\sqrt{2}$

Explanation:

To solve this problem, use the Pythagorean theorem: the sum of the square of the legs equals the square of the hyoptenuse or, mathematically, a^2+b^2=c^2 .

Plug in our given values.

a^2+b^2=c^2

4^2+4^2=c^2

16+16=c^2

32=c^2

$\sqrt{32}=\sqrt{c^2}$

is not a perfect square, but let's see if we can find a factor that is a perfect square.

$\sqrt{16*2}=\sqrt{c^2}$

IS a perfect square, so we can simplify!

$\sqrt{16}*\sqrt{2}=\sqrt{c^2}$

$4*\sqrt2=c$

$4\sqrt{2}=c$

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

Solve for :

Question_12

Possible Answers:

x=12

$x=6\sqrt2$

$x=9\sqrt2$

x=9

Correct answer:

$x=6\sqrt2$

Explanation:

Solve for using the Pythagorean Theorem:

c^2=a^2+b^2

x=c

x^2=6^2+6^2=36+36=72

$\sqrt{x^2}=\sqrt{72}$

$x=\sqrt{36}\times \sqrt2$

$x=6\sqrt2$

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High School Math : Plane Geometry

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #18 : How To Find The Length Of The Side Of A Right Triangle

Example Question #14 : How To Find The Length Of The Side Of A Right Triangle

Example Question #19 : How To Find The Length Of The Side Of A Right Triangle

Example Question #21 : 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 #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #1 : 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

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

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

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