How to find the length of the side of an equilateral triangle - Intermediate Geometry

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Question

The area of an equilateral triangle is $16\sqrt3$ , what is the length of each side?

Answer

An equilateral triangle can be broken down into 2 30-60-90 right triangles (see image). The length of a side (the base) is 2x while the length of the height is $x\sqrt3$ . The area of a triangle can be calculated using the following equation:

$A=\frac{1}{2}bh$

Therefore, if equals the length of a side:

$A=16\sqrt3=\frac{1}{2}bh=\frac{1}{2}(2x)(x\sqrt3)=x^2\sqrt3$

$16\sqrt3=x^2\sqrt3$

A length of the side equals 2x:

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Question

What is the area of this triangle if ?

Answer

We know the formula for the area of an equilateral triangle is:

$Area=\frac{\sqrt{3}}{4}a^2$

if is the side of the triangle.

So, since we are told that , we can substitute in for and solve for the area of the triangle:

$Area=\frac{\sqrt{3}}{4}1^2=\frac{\sqrt{3}}{4}1=\frac{\sqrt{3}}{4}:cm^2$

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Question

Find if the perimeter of this triangle is .

Answer

This triangle is equilateral; we can tell because each of its sides are the same length, . To find the length of one side, we need to divide the perimeter by :

$a=\frac{18}{3}=6$

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Question

What is side if the perimeter of this triangle is $\frac{3}{7}:cm$ ?

Answer

Since each of this triangle's sides is equal in length, it is equilateral. To find the length of one side of an equilateral triangle, we need to divide the perimeter by .

$a=\frac{\frac{3}{7}}{3}=\frac{1}{7}:cm$

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Question

The height of the triangle is feet.

What is the length of the base of the triangle to the nearest tenth?

Answer

Since it is an equilateral triangle, the line that represents the height bisects it into a 30-60-90 triangle.

Here you may use $sin(60)=\frac{4}{hypotenuse}$ and solve for hypotenuse to find one of the sides of the triangle.

Use the definition of an equilateral triangle to know that the answer of the hypotenuse also applies to the base of the triangle.

Therefore,

$hypotenuse=\frac{4}{sin(60)}=4.6$

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Question

The height of an equilateral triangle is 5. How long are its sides?

Answer

The height of an equilateral triangle, shown by the dotted line, is also one of the legs of a right triangle:

The hypotenuse is x, the length of each side in this equilateral triangle, and then the other leg is half of that, 0.5x.

To solve for x, use Pythagorean Theorem:

square the terms on the left

combine like terms by subtracting 0.25 x squared from both sides

divide both sides by 0.75

$33. \overline 3 = x^2$ take the square root of both sides

$\sqrt{ 33. \overline 3 } = x$

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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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.

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

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

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

$\text{Perimeter}=5(\frac{4\sqrt3}{3})=\frac{20\sqrt3}{3}=11.55$

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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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.

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

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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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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.

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

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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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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.

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

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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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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.

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

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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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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.

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

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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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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.

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

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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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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.

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

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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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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.

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

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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Question

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

Find the perimeter of the shape.

Answer

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 triangles.

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

Recall that a 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.

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

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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Question

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

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

Answer

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

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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Question

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

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

Answer

Assume $m \angle A = 60^{\circ }$ . Then, since , 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, , and the perimeter, or the sum of the lengths of the sides, is

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

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