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Intermediate Geometry : Triangles

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Example Question #291 : Triangles

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

82.33

80.09

91.37

86.59

Correct answer:

86.59

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

Recall that the side lengths in a 30-60-90 triangle are in a $1:\sqrt3:2$ ratio. Thus, the radius of the circle, which is also the base of the triangle, the height of the triangle, and the side length of the triangle are in the same ratio.

We can then set up the following to determine the length of the side of the equilateral triangle:

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

Rearrange the equation to solve for the length of the side.

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

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

$\text{side}=\frac{2\sqrt3(32)}{3}=\frac{42\sqrt3}{3}$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(\frac{42\sqrt3}{3})+\frac{\pi(\frac{42\sqrt3}{3})}{2}=\frac{84\sqrt3}{3}+\frac{21\pi\sqrt3}{3}=86.59$

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Example Question #91 : Equilateral Triangles

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

94.18

92.20

96.77

98.96

Correct answer:

98.96

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

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

Rearrange the equation to solve for the length of the side.

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

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

$\text{side}=\frac{2\sqrt3(24)}{3}=16\sqrt3$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(16\sqrt3)+\frac{\pi(16\sqrt3)}{2}=32\sqrt3+8\pi\sqrt3=98.96$

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Example Question #291 : Triangles

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

125.20

106.20

110.20

107.20

Correct answer:

107.20

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

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

Rearrange the equation to solve for the length of the side.

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

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

$\text{side}=\frac{2\sqrt3(26)}{3}=\frac{52\sqrt3}{3}$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(\frac{52\sqrt3}{3})+\frac{\pi(\frac{52\sqrt3}{3})}{2}=\frac{104\sqrt3}{3}+\frac{26\pi\sqrt3}{3}=107.20$

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Example Question #93 : Equilateral Triangles

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

111.87

110.29

117.99

111.33

Correct answer:

111.33

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

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

Rearrange the equation to solve for the length of the side.

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

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

$\text{side}=\frac{2\sqrt3(27)}{3}=18\sqrt3$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(18\sqrt3)+\frac{\pi(18\sqrt3)}{2}=36\sqrt3+9\pi\sqrt3=111.33$

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Example Question #94 : Equilateral Triangles

If an equilateral triangle has a height of $13.7\;cm$ , what would be the perimeter? Round to the nearest tenth.

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Possible Answers:

$19.4\;cm$

$58.2\;cm$

$47.4\;cm$

$41.1\;cm$

$37.7\;cm$

Correct answer:

$47.4\;cm$

Explanation:

Since the triangle is equilateral, all sides are the same length. Therefore, we only need to find the length of one side to determine the perimeter. We can do this by means of the Pythagorean Theorem. In the attached figure, the equilateral triangle has been divided into two right triangles, for which the Pythagorean Theorem can be performed:

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With representing the length of one side, we can solve for using the Pythagorean Theorem:

$\\a^2=(13.7\;cm)^2+(\frac{1}{2}a)^2\\a^2=187.7\;cm^2+\frac{1}{4}a^2\\ \frac{3}{4}a^2=187.7\;cm^2\\a^2=250.3\;cm^2$

$a=\sqrt{250.3\;cm^2}\approx15.8\;cm$

Now that we know the length of one side, we can solve for the total perimeter by summing like sides:

$Perimeter=15.8\;cm+15.8\;cm+15.8\;cm=47.4\;cm$

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Example Question #291 : Triangles

Given: $\bigtriangleup ABC$ ; $\overline{AB}$ has length nine inches.

True or false: The perimeter of $\bigtriangleup ABC$ is one yard.

Possible Answers:

False

True

Correct answer:

False

Explanation:

The perimeter of an equilateral triangle - one with three sides of equal length - is equal to three times the length of one side. Therefore, $\bigtriangleup ABC$ has perimeter

$9 \textrm{ in }\times 3 = 27 \textrm{ in }$

One yard is equal to 36 inches, making the statement false.

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Example Question #96 : Equilateral Triangles

Locate and , the midpoints of sides $\overline{AB}$ and $\overline{AC}$ of an equilateral triangle. MN = 12 .

Give the perimeter of the triangle.

Possible Answers:

108

144

Correct answer:

Explanation:

A midsegment of a triangle - a segment whose endpoints are the midpoints of two sides - is, by the Triangle Midsegment Theorem, parallel to the third side, and is half the length of that side. Therefore,

$MN = \frac{1}{2} \cdot BC$

Substitute and solve:

$12= \frac{1}{2} \cdot BC$

$12 \cdot 2 = \frac{1}{2} \cdot BC \cdot 2$

24 =BC

Since the lengths of the sides of an equilateral triangle are equal, the perimeter is three times this, or

$3 \cdot 24 = 72$

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Intermediate Geometry : Triangles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #291 : Triangles

Example Question #91 : Equilateral Triangles

Example Question #291 : Triangles

Example Question #93 : Equilateral Triangles

Example Question #94 : Equilateral Triangles

Example Question #291 : Triangles

Example Question #96 : Equilateral Triangles

All Intermediate Geometry Resources

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