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Basic Geometry : How to find the perimeter of a 45/45/90 right isosceles triangle

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Example Question #161 : Triangles

A right isosceles triangle is stacked on top of a square as shown in the figure. Find the perimeter of the compound shape.

Possible Answers:

81.09

88.28

85.61

97.82

Correct answer:

88.28

Explanation:

Notice that the hypotenuse of the right isosceles triangle is also the length of a side of the square.

First, we will need to find the length of the legs of the triangle by using the Pythagorean theorem.

$\text{leg}^2+\text{leg}^2=\text{side}^2$

$2(\text{leg})^2=\text{side}^2$

$\text{leg}^2=\frac{\text{side}^2}{2}$

$\text{leg}=\sqrt{\frac{\text{side}^2}{2}}=\frac{\text{side}\sqrt2}{2}$

Substitute in the given length of the hypotenuse to find the length of the leg of the triangle.

$\text{leg}=\frac{20\sqrt2}{2}$

$\text{leg}=10\sqrt2$

In order to find the perimeter, add up the lengths outlined in red. The perimeter includes the two legs of the triangle and three sides of the square.

Therefore:

$\text{Perimeter}=20+20+20+10\sqrt2+10\sqrt2$

$\text{Perimeter}=88.28$

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Example Question #162 : Triangles

A right isosceles triangle is stacked on top of a square as shown in the figure. Find the perimeter of the compound shape.

Possible Answers:

97.11

102.58

99.42

95.10

Correct answer:

97.11

Explanation:

Notice that the hypotenuse of the right isosceles triangle is also the length of a side of the square.

First, we will need to find the length of the legs of the triangle by using the Pythagorean theorem.

$\text{leg}^2+\text{leg}^2=\text{side}^2$

$2(\text{leg})^2=\text{side}^2$

$\text{leg}^2=\frac{\text{side}^2}{2}$

$\text{leg}=\sqrt{\frac{\text{side}^2}{2}}=\frac{\text{side}\sqrt2}{2}$

Substitute in the given length of the hypotenuse to find the length of the leg of the triangle.

$\text{leg}=\frac{22\sqrt2}{2}$

$\text{leg}=11\sqrt2$

In order to find the perimeter, add up the lengths outlined in red. The perimeter includes the two legs of the triangle and three sides of the square.

Therefore:

$\text{Perimeter}=22+22+22+11\sqrt2+11\sqrt2$

$\text{Perimeter}=97.11$

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Example Question #42 : How To Find The Perimeter Of A 45/45/90 Right Isosceles Triangle

A right isosceles triangle is stacked on top of a square as shown in the figure. Find the perimeter of the compound shape.

Possible Answers:

101.09

105.94

107.28

102.42

Correct answer:

105.94

Explanation:

Notice that the hypotenuse of the right isosceles triangle is also the length of a side of the square.

First, we will need to find the length of the legs of the triangle by using the Pythagorean theorem.

$\text{leg}^2+\text{leg}^2=\text{side}^2$

$2(\text{leg})^2=\text{side}^2$

$\text{leg}^2=\frac{\text{side}^2}{2}$

$\text{leg}=\sqrt{\frac{\text{side}^2}{2}}=\frac{\text{side}\sqrt2}{2}$

Substitute in the given length of the hypotenuse to find the length of the leg of the triangle.

$\text{leg}=\frac{24\sqrt2}{2}$

$\text{leg}=12\sqrt2$

In order to find the perimeter, add up the lengths outlined in red. The perimeter includes the two legs of the triangle and three sides of the square.

Therefore:

$\text{Perimeter}=24+24+24+12\sqrt2+12\sqrt2$

$\text{Perimeter}=105.94$

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Basic Geometry : How to find the perimeter of a 45/45/90 right isosceles triangle

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #161 : Triangles

Example Question #162 : Triangles

Example Question #42 : How To Find The Perimeter Of A 45/45/90 Right Isosceles Triangle

All Basic Geometry Resources

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