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High School Math : Equilateral Triangles

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Example Questions

Example Question #712 : Intermediate Geometry

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

Possible Answers:

$4\sqrt{3}$

$6\sqrt{2}$

$4\sqrt{2}$

$6\sqrt{3}$

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 #482 : Plane Geometry

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

200

180

208

188

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

Determine the perimeter of the following equilateral triangle:

Screen_shot_2014-02-27_at_6.40.28_pm

Possible Answers:

$24\sqrt{3}cm$

24cm

$16\sqrt{3}cm$

64cm

16cm

Correct answer:

24cm

Explanation:

The formula for the perimeter of an equilateral triangle is:

P=3(s) ,

where is the length of the side.

Plugging in our value, we get:

P=3(8cm)=24cm

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

An equilateral triangle has a side of length feet. What is the perimeter of the triangle?

Possible Answers:

feet

It cannot be determined from the information given.

foot

Correct answer:

feet

Explanation:

An equilateral triangle by definition. has three congruent sides. Thus, if one side is feet long, then all three sides are each feet long. We also know that the perimeter is the sum of all sides. Therefore, our perimeter is 2 + 2 + 2 = 6

Since we were given units of feet, our final answer is feet.

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High School Math : Equilateral Triangles

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #712 : Intermediate Geometry

Example Question #482 : Plane Geometry

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

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

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

All High School Math Resources

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