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High School Math : Plane Geometry

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Example Questions

Example Question #481 : Geometry

Equilateral_triangle

An equilateral triangle has a side length of s=4 cm . What is its height, ?

Possible Answers:

3.46cm

5.19cm

1.73cm

Not enough information to solve

4.36cm

Correct answer:

3.46cm

Explanation:

An altitude slices an equilateral triangle into two $30^{\circ}-60^{\circ}-90^{\circ}$ triangles. These triangles follow a side-length pattern. The smallest of the two legs equals and the hypotenuse equals . By way of the Pythagorean Theorem, the longest leg or $h=x\sqrt{3}$ .

Therefore, we can find the height of the altitude of this triangle by designating a value to . The hypotenuse of one of the $30^{\circ}-60^{\circ}-90^{\circ}$ is also the side of the original equilateral triangle. Therefore, one can say that 2x=s=4cm and x=2cm .

$h=x\sqrt{3}$

$h=2\sqrt{3}cm$

$\rightarrow3.46cm$

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Example Question #1 : How To Find The Height Of An Equilateral Triangle

What is the height of an equilateral triangle with side 6?

Possible Answers:

$3\sqrt{3}$

$6\sqrt{2}$

Correct answer:

$3\sqrt{3}$

Explanation:

When you draw the height in an equilateral triangle, it makes two 30-60-90 triangles. Because of that relationship, the height (which is across from the $60^{\circ}$ ) is $3\sqrt{3}$ .

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Example Question #483 : Plane Geometry

Find the height of the following equilateral triangle:

Triangle

Possible Answers:

20m

$20\sqrt{3}m$

10m

$10\sqrt{3}m$

$15\sqrt{3}m$

Correct answer:

$10\sqrt{3}m$

Explanation:

Each angle in an equilateral triangle is $60^{\circ}$ .

Use the formula for 30-60-90 triangles in order to find the length of the height.

The formula is:

$a-a\sqrt{3}-2a$

Where is the length of the side opposite the $\measuredangle 30$

If we were to create a 30-60-90 triangle by drawing the height, the length of the side is 20m , the base is 10m , and the height is $10\sqrt{3}m$ .

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Example Question #4 : How To Find The Height Of An Equilateral Triangle

Solve for the value of X in the following equilateral triangle:

Screen_shot_2014-02-27_at_6.35.43_pm

Possible Answers:

$6\sqrt{3}m$

$3\sqrt{3}m$

$9\sqrt{3}m$

Correct answer:

$6\sqrt{3}m$

Explanation:

If we draw a line segment between X and the base of the triangle, we form a 30-60-90 triangle.

We can use the relationships between the sides of a 30-60-90 triangle in order to find the length of X.

We know the base opposite the $\measuredangle60$ is .

The value of the height opposite the $\measuredangle30$ must then be $\frac{9m\sqrt{3}}{3}$ , or $3\sqrt{3}m$ .

Therefore, the value of X will be twice the value of the height:

$X = 2(3\sqrt{3}m)=6\sqrt{3}m$

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Example Question #1 : How To Find The Height Of An Equilateral Triangle

What is the height of an equilateral triangle with a side length of 8 in?

Possible Answers:

$6\sqrt{3}$

$4\sqrt{3}$

$4\sqrt{2}$

$6\sqrt{2}$

Correct answer:

$4\sqrt{3}$

Explanation:

An equilateral triangle has three congruent sides, and is also an equiangular triangle with three congruent angles that each meansure 60 degrees.

To find the height we divide the triangle into two special 30 - 60 - 90 right triangles by drawing a line from one corner to the center of the opposite side. This segment will be the height, and will be opposite from one of the 60 degree angles and adjacent to a 30 degree angle. The special right triangle gives side ratios of , $x\sqrt{3}$ , and . The hypoteneuse, the side opposite the 90 degree angle, is the full length of one side of the triangle and is equal to . Using this information, we can find the lengths of each side fo the special triangle.

$8=2x\rightarrow 4=x\rightarrow 4\sqrt{3}=x\sqrt{3}$

The side with length $x\sqrt{3}$ will be the height (opposite the 60 degree angle). The height is $4\sqrt{3}$ inches.

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Example Question #1 : How To Find The Perimeter Of An Equilateral Triangle

Gre11

A square rug border consists of a continuous pattern of equilateral triangles, with isosceles triangles as corners, one of which is shown above. If the length of each equilateral triangle side is 5 inches, and there are 40 triangles in total, what is the total perimeter of the rug?

The inner angles of the corner triangles is 30°.

Possible Answers:

124

188

208

180

200

Correct answer:

188

Explanation:

There are 2 components to this problem. The first, and easier one, is recognizing how much of the perimeter the equilateral triangles take up—since there are 40 triangles in total, there must be 40 – 4 = 36 of these triangles. By observation, each contributes only 1 side to the overall perimeter, thus we can simply multiply 36(5) = 180" contribution.

The second component is the corner triangles—recognizing that the congruent sides are adjacent to the 5-inch equilateral triangles, and the congruent angles can be found by

180 = 30+2x → x = 75°

We can use ratios to find the unknown side:

75/5 = 30/y → 75y = 150 → y = 2''.

Since there are 4 corners to the square rug, 2(4) = 8'' contribution to the total perimeter. Adding the 2 components, we get 180+8 = 188 inch perimeter.

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Example Question #2 : How To Find The Perimeter Of An Equilateral Triangle

The height of an equilateral triangle is $\dpi{100} \small 2\sqrt{3}$

What is the triangle's perimeter?

Possible Answers:

$\dpi{100} \small 2\sqrt{2}$

Correct answer:

Explanation:

An altitude drawn in an equilateral triangle will form two 30-60-90 triangles. The height of equilateral triangle is the length of the longer leg of the 30-60-90 triangle. The length of the equilateral triangle's side is the length of the hypotenuse of the 30-60-90.

The ratio of the length of the hypotenuse to the length of the longer leg of a 30-60-90 triangle is $\dpi{100} \small 2:\sqrt{3}$

The length of the longer leg of the 30-60-90 triangle in this problem is $\dpi{100} \small 2\sqrt{3}$

Using this ratio, we find that the length of this triangle's hypotenuse is 4. Thus the perimeter of the equilateral triangle will be 4 multiplied by 3, which is 12.

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Example Question #21 : Equilateral Triangles

Equilateral_triangle

An equilateral triangle has a side length of $s=\sqrt{3}$ . What is its perimeter?

Possible Answers:

Not enough information to solve.

3.46

2.50

5.20

6.43

Correct answer:

5.20

Explanation:

An equilateral triangle possesses three sides of equal lengths. Therefore, we can easily calculate its perimeter by tripling the given side length.

P=3*s

$P=3*\sqrt{3}$

$\rightarrow 5.20$

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Example Question #22 : Equilateral Triangles

Equilateral_triangle

An equilateral triangle has an altitude length of $h=2\sqrt{3}$ . What is its perimeter?

Possible Answers:

10.39

13.09

Not enough information to solve

Correct answer:

Explanation:

An altitude slices an equilateral triangle into two $30^{\circ}-60^{\circ}-90^{\circ}$ triangles. These triangles follow a side length pattern. The smallest of the two legs equals and the hypotenuse equals . By way of the Pythagorean Theorem, the longest leg or $h=x\sqrt{3}$ .

We have the length of the altitude of the triangle $h=\sqrt{3}$ . We can solve for the smallest side via substitution and simple algebra.

$h=x\sqrt{3}$

$2\sqrt{3}=x\sqrt{3}$

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

x=2

Note, this is only the smallest side of one of the $30^{\circ}-60^{\circ}-90^{\circ}$ triangles. It needs to be doubled to equal a complete side of the equilateral triangle.

s=2x

s=4

Now, the side length can be tripled to calculate the perimeter.

P=3s

P=3*4

$\rightarrow12$

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Example Question #1 : How To Find The Perimeter Of An Equilateral Triangle

Find the perimeter of the following equilateral triangle:

Triangle

Possible Answers:

40m

20m

60m

120m

80m

Correct answer:

60m

Explanation:

The formula for the perimeter of an equilateral triangle is:

P=3(s)

Where is the length of the side

Plugging in our values, we get:

P=3(20m)=60m

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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 #481 : Geometry

Example Question #1 : How To Find The Height Of An Equilateral Triangle

Example Question #483 : Plane Geometry

Example Question #4 : How To Find The Height Of An Equilateral Triangle

Example Question #1 : How To Find The Height Of An Equilateral Triangle

Example Question #1 : How To Find The Perimeter Of An Equilateral Triangle

Example Question #2 : How To Find The Perimeter Of An Equilateral Triangle

Example Question #21 : Equilateral Triangles

Example Question #22 : Equilateral Triangles

Example Question #1 : How To Find The Perimeter Of An Equilateral Triangle

All High School Math Resources

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