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Intermediate Geometry : Equilateral Triangles

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Example Questions

Example Question #11 : Equilateral Triangles

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

28.76

34.64

29.61

31.20

Correct answer:

34.64

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into congruent 30-60-90 triangles.

We can then use the height to find the length of the side of the triangle.

Recall that a 30-60-90 triangle has sides that are in ratios of $1:\sqrt3:2$ . The smallest side in the given figure is the base, the second longest side is the height, and the longest side is the side of the triangle itself.

Thus, we can use the ratio and the length of the height to set up the following equation:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the given height to find the length of the side.

$\text{side}=\frac{2\sqrt3(6)}{3}=4\sqrt3$

Now, since the perimeter of the shape consists of of these sides, we can use the following equation to find the perimeter.

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(4\sqrt3)=20\sqrt3=34.64$

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Example Question #211 : Triangles

An equilateral triangle is placed on a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

42.51

46.19

47.99

41.24

Correct answer:

46.19

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into congruent 30-60-90 triangles.

We can then use the height to find the length of the side of the triangle.

Thus, we can use the ratio and the length of the height to set up the following equation:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the given height to find the length of the side.

$\text{side}=\frac{2\sqrt3(8)}{3}=\frac{16\sqrt3}{3}$

Now, since the perimeter of the shape consists of of these sides, we can use the following equation to find the perimeter.

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(\frac{16\sqrt3}{3})=\frac{80\sqrt3}{3}=46.19$

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Example Question #13 : How To Find The Length Of The Side Of An Equilateral Triangle

An equilateral triangle is placed on a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

53.53

52.09

51.85

51.96

Correct answer:

51.96

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into congruent 30-60-90 triangles.

We can then use the height to find the length of the side of the triangle.

Thus, we can use the ratio and the length of the height to set up the following equation:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the given height to find the length of the side.

$\text{side}=\frac{2\sqrt3(9)}{3}=6\sqrt3$

Now, since the perimeter of the shape consists of of these sides, we can use the following equation to find the perimeter.

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(6\sqrt3)=30\sqrt3=51.96$

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Example Question #14 : How To Find The Length Of The Side Of An Equilateral Triangle

An equilateral triangle is placed on a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

58.99

57.74

53.35

58.12

Correct answer:

57.74

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into congruent 30-60-90 triangles.

We can then use the height to find the length of the side of the triangle.

Thus, we can use the ratio and the length of the height to set up the following equation:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the given height to find the length of the side.

$\text{side}=\frac{2\sqrt3(10)}{3}=\frac{20\sqrt3}{3}$

Now, since the perimeter of the shape consists of of these sides, we can use the following equation to find the perimeter.

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(\frac{20\sqrt3}{3})=\frac{100\sqrt3}{3}=57.74$

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Example Question #15 : How To Find The Length Of The Side Of An Equilateral Triangle

An equilateral triangle is placed on a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

77.54

69.28

71.50

76.29

Correct answer:

69.28

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into congruent 30-60-90 triangles.

We can then use the height to find the length of the side of the triangle.

Thus, we can use the ratio and the length of the height to set up the following equation:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the given height to find the length of the side.

$\text{side}=\frac{2\sqrt3(12)}{3}=8\sqrt3$

Now, since the perimeter of the shape consists of of these sides, we can use the following equation to find the perimeter.

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(8\sqrt3)=40\sqrt3=69.28$

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Example Question #16 : How To Find The Length Of The Side Of An Equilateral Triangle

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

87.94

88.40

85.11

86.60

Correct answer:

86.60

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into congruent 30-60-90 triangles.

We can then use the height to find the length of the side of the triangle.

Thus, we can use the ratio and the length of the height to set up the following equation:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the given height to find the length of the side.

$\text{side}=\frac{2\sqrt3(15)}{3}=10\sqrt3$

Now, since the perimeter of the shape consists of of these sides, we can use the following equation to find the perimeter.

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(10\sqrt3)=50\sqrt3=86.60$

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Example Question #17 : How To Find The Length Of The Side Of An Equilateral Triangle

An equilateral triangle is placed on a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

93.49

90.04

100.08

92.38

Correct answer:

92.38

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into congruent 30-60-90 triangles.

We can then use the height to find the length of the side of the triangle.

Thus, we can use the ratio and the length of the height to set up the following equation:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the given height to find the length of the side.

$\text{side}=\frac{2\sqrt3(16)}{3}=\frac{32\sqrt3}{3}$

Now, since the perimeter of the shape consists of of these sides, we can use the following equation to find the perimeter.

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(\frac{32\sqrt3}{3})=\frac{160\sqrt3}{3}=92.38$

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Example Question #18 : How To Find The Length Of The Side Of An Equilateral Triangle

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

100.82

107.62

103.92

99.46

Correct answer:

103.92

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into congruent 30-60-90 triangles.

We can then use the height to find the length of the side of the triangle.

Thus, we can use the ratio and the length of the height to set up the following equation:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the given height to find the length of the side.

$\text{side}=\frac{2\sqrt3(18)}{3}=12\sqrt3$

Now, since the perimeter of the shape consists of of these sides, we can use the following equation to find the perimeter.

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(12\sqrt3)=60\sqrt3=103.92$

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Example Question #19 : How To Find The Length Of The Side Of An Equilateral Triangle

Given: Regular Pentagon PENTA with center . Construct segments $\overline{CT}$ and $\overline{CN}$ to form $\bigtriangleup CTN$ .

True or false: $\bigtriangleup CTN$ is an equilateral triangle.

Possible Answers:

True

False

Correct answer:

False

Explanation:

Below is regular Pentagon PENTA with center , a segment drawn from to each vertex - that is, each of its radii drawn.

Pentagon a

The measure of each angle of a regular pentagon can be calculated by setting equal to 5 in the formula

$\frac{(N-2)180^{\circ }}{N}$

and evaluating:

$\frac{(5-2)180^{\circ }}{5} = \frac{3 \cdot 180^{\circ }}{5} = \frac{540^{\circ }}{5} = 108^{\circ }$

By symmetry, each radius bisects one of these angles. Specifically,

$m \angle CTN = \frac{1}{2} m \angle ATN = \frac{1}{2} \cdot 108^{\circ } = 54 ^{\circ }$ .

An equilateral triangle has three angles of measure $60^{\circ }$ , so $\bigtriangleup CTN$ is not equilateral.

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Example Question #663 : Intermediate Geometry

Equilateral

Refer to the above diagram. $\bigtriangleup ABC$ has perimeter 56.

True or false: $m \angle A = 60^{\circ }$

Possible Answers:

False

True

Correct answer:

False

Explanation:

Assume $m \angle A = 60^{\circ }$ . Then, since AB = AC , it follows by the Isosceles Triangle Theorem that their opposite angles are also congruent. Since the measures of the angles of a triangle total $180^{\circ }$ , letting $t = m \angle B = m \angle C$ :

$m \angle A + m \angle B + m \angle C = 180^{\circ }$

$60^{\circ }+ t+t = 180^{\circ }$

$60^{\circ }+2t = 180^{\circ }$

$60^{\circ }+2t - 60^{\circ } = 180^{\circ } - 60^{\circ }$

$2t = 120^{\circ }$

$\frac{2t }{2}= \frac{120^{\circ }}{2}$

$t = 60^{\circ }$

All three angles have measure $60^{\circ }$ , making $\bigtriangleup ABC$ equiangular and, as a consequence, equilateral. Therefore, BC = AB = AC = 18 , and the perimeter, or the sum of the lengths of the sides, is

P= AB + BC + AC = 18 + 18+18 = 54

However, the perimeter is given to be 56. Therefore, $m \angle A \ne 60^{\circ }$ .

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Intermediate Geometry : Equilateral Triangles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #11 : Equilateral Triangles

Example Question #211 : Triangles

Example Question #13 : How To Find The Length Of The Side Of An Equilateral Triangle

Example Question #14 : How To Find The Length Of The Side Of An Equilateral Triangle

Example Question #15 : How To Find The Length Of The Side Of An Equilateral Triangle

Example Question #16 : How To Find The Length Of The Side Of An Equilateral Triangle

Example Question #17 : How To Find The Length Of The Side Of An Equilateral Triangle

Example Question #18 : How To Find The Length Of The Side Of An Equilateral Triangle

Example Question #19 : How To Find The Length Of The Side Of An Equilateral Triangle

Example Question #663 : Intermediate Geometry

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