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Common Core: 8th Grade Math : Geometry

Study concepts, example questions & explanations for Common Core: 8th Grade Math

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All Common Core: 8th Grade Math Resources

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Example Questions

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

If a right triangle has a base of and a height of , what is the length of the hypotenuse?

Possible Answers:

$\sqrt{11}$

$\sqrt{65}$

$\sqrt{14}$

Correct answer:

$\sqrt{65}$

Explanation:

To solve this problem, we must utilize the Pythagorean Theorom, which states that:

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

We know that the base is , so we can substitute in for . We also know that the height is , so we can substitute in for .

$7^{2}+4^{2}=c^{2}$

Next we evaluate the exponents:

$7^{2}=49$

$4^{2}=16$

Now we add them together:

49+16=65

Then, $65=c^{2}$ .

is not a perfect square, so we simply write the square root as $\sqrt{65}$ .

$c=\sqrt{65}$

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Example Question #101 : Geometry

If a right triangle has a base of and a height of , what is the length of the hypotenuse?

Possible Answers:

144

$\sqrt{90}$

Correct answer:

$\sqrt{90}$

Explanation:

To solve this problem, we are going to use the Pythagorean Theorom, which states that $a^{2}+b^{2}=c^{2}$ .

We know that this particular right triangle has a base of , which can be substituted for , and a height of , which can be substituted for . If we rewrite the theorom using these numbers, we get:

$3^{2}+9^{2}=c^{2}$

Next, we evaluate the expoenents:

$3^{2}=9$

$9^{2}=81$

$9+81=c^{2}$

9+81=90

Then, $90=c^{2}$ .

To solve for , we must find the square root of . Since this is not a perfect square, our answer is simply $c=\sqrt{90}$ .

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

What is the hypotenuse of a right triangle with sides 5 and 8?

Possible Answers:

100

$\sqrt{89}$

undefined

Correct answer:

$\sqrt{89}$

Explanation:

According to the Pythagorean Theorem, the equation for the hypotenuse of a right triangle is $a^{2} + b^{2}=c^{2}$ . Plugging in the sides, we get $5^{2}+8^{2}=c^{2}$ . Solving for , we find that the hypotenuse is $\sqrt{89}$ :

25+64=c^2

89=c^2

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Example Question #3 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

In a right triangle, two sides have length . Give the length of the hypotenuse in terms of .

Possible Answers:

$2\sqrt{3}t$

$\sqrt{2}t$

$2\sqrt{2}t$

$\sqrt{3}t$

Correct answer:

$2\sqrt{2}t$

Explanation:

By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let hypotenuse and side length.

$c^2=s^2+s^2\Rightarrow c^2=(2t)^2+(2t)^2\Rightarrow c^2=8t^2\Rightarrow c=\sqrt{8t^2}=2\sqrt{2}t$

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

In a right triangle, two sides have lengths 5 centimeters and 12 centimeters. Give the length of the hypotenuse.

Possible Answers:

$13\ cm$

$14\ cm$

$15\ cm$

$14.5\ cm$

$13.5\ cm$

Correct answer:

$13\ cm$

Explanation:

This triangle has two angles of 45 and 90 degrees, so the third angle must measure 45 degrees; this is therefore an isosceles right triangle.

By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let hypotenuse and , lengths of the other two sides.

$c^2=a^2+b^2\Rightarrow c^2=5^2+12^2\Rightarrow c^2=25+144=169$

$\Rightarrow c=\sqrt{169}\Rightarrow c=13\ cm$

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

In a right triangle, the legs are 7 feet long and 12 feet long. How long is the hypotenuse?

Possible Answers:

193

$3\sqrt{19$

$\sqrt{193}$

Correct answer:

$\sqrt{193}$

Explanation:

The pythagorean theory should be used to solve this problem.

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

$7^{2}+12^{2}+=c^{2}$

$49+144=c^{2}$

$193=c^{2}$

$\sqrt{193}=c$

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Example Question #62 : Plane Geometry

The legs of a right triangle are equal to 4 and 5. What is the length of the hypotenuse?

Possible Answers:

$\sqrt{41}$

Correct answer:

$\sqrt{41}$

Explanation:

If the legs of a right triangle are 4 and 5, to find the hypotenuse, the following equation must be used to find the hypotenuse, in which c is equal to the hypotenuse:

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

$4^2{+5^{2}}=c^2$

16+25=c^2

$41=c^{2}$

$c={\sqrt{41}}$

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Example Question #486 : Geometry

Righttriangle

Triangle ABC is a right triangle. If the length of side A = 3 inches and C = 5 inches, what is the length of side B?

Possible Answers:

1/2 inches

1 inches

6 inches

4 inches

4.5 inches

Correct answer:

4 inches

Explanation:

Using the Pythagorean Theorem, we know that $a^{2} + b^{2} = c^{2}$ .

This gives:

$3^{2} + b^{2} = 5^{2}$

$9 + b^{2} = 25$

Subtracting 9 from both sides of the equation gives:

$b^{2} = 16$

b = 4 inches

Righttriangle

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Example Question #102 : Geometry

An 8-foot-tall tree is perpendicular to the ground and casts a 6-foot shadow. What is the distance, to the nearest foot, from the top of the tree to the end of the shadow?

Possible Answers:

$\dpi{100} \small 4$

$\dpi{100} \small 5$

$\dpi{100} \small 10$

$\dpi{100} \small 8$

$\dpi{100} \small 6$

Correct answer:

$\dpi{100} \small 10$

Explanation:

In order to find the distance from the top of the tree to the end of the shadow, draw a right triangle with the height(tree) labeled as 8 and base(shadow) labeled as 6:

Screen_shot_2013-08-16_at_12.34.40_am

From this diagram, you can see that the distance being asked for is the hypotenuse. From here, you can either use the Pythagorean Theorem:

$\dpi{100} \small a^{2}+b^{2}=c^{2}$

or you can notice that this is simililar to a 3-4-5 triangle. Since the lengths are just increased by a factor of 2, the hypotenuse that is normally 5 would be 10.

8^2+6^2=c^2

64+36=c^2

100=c^2

10=c

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Example Question #72 : Right Triangles

Find the perimeter of the polygon.

Possible Answers:

Correct answer:

Explanation:

Divide the shape into a rectangle and a right triangle as indicated below.

Find the hypotenuse of the right triangle with the Pythagorean Theorem, a^2 + b^2 = c^2 , where and are the legs of the triangle and is its hypotenuse.

(8)^2+(6)^2 = c^2

100 = c^2

$\sqrt{100} = \sqrt{c^2}$

c =10

This is our missing length.

Now add the sides of the polygon together to find the perimeter:

20 + 10 + 26 + 8 = 64

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Common Core: 8th Grade Math : Geometry

Study concepts, example questions & explanations for Common Core: 8th Grade Math

All Common Core: 8th Grade Math Resources

Example Questions

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

Example Question #101 : Geometry

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

Example Question #3 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

Example Question #3 : 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 #62 : Plane Geometry

Example Question #486 : Geometry

Example Question #102 : Geometry

Example Question #72 : Right Triangles

All Common Core: 8th Grade Math Resources

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