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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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8 Diagnostic Tests 250 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #8 : 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:

14.28

19.04

18.22

17.32

Correct answer:

17.32

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.

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

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

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

$\text{Perimeter}=5(2\sqrt3)=10\sqrt3=17.32$

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Example Question #9 : 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:

23.09

23.18

25.66

21.01

Correct answer:

23.09

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.

$\text{side}=\frac{2\sqrt3(4)}{3}=\frac{8\sqrt3}{3}$

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

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

$\text{Perimeter}=5(\frac{8\sqrt3}{3})=\frac{40\sqrt3}{3}=23.09$

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Example Question #1 : 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:

28.12

28.87

29.11

27.41

Correct answer:

28.87

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.

$\text{side}=\frac{2\sqrt3(5)}{3}=\frac{10\sqrt3}{3}$

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

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

$\text{Perimeter}=5(\frac{10\sqrt3}{3})=\frac{50\sqrt3}{3}=28.87$

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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.

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.

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

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

Report an Error

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.

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

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

Report an Error

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.

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

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

Report an Error

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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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #8 : How To Find The Length Of The Side Of An Equilateral Triangle

Example Question #9 : How To Find The Length Of The Side Of An Equilateral Triangle

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

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

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