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Basic Geometry : How to find the length of the hypotenuse of a 45/45/90 right isosceles triangle : Pythagorean Theorem

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Example Question #51 : 45/45/90 Right Isosceles Triangles

A rectangle is inscribed in a right isosceles triangle. The length of the rectangle is half the hypotenuse of the triangle and is shown in the provided figure. What is the area of the shaded region?

Possible Answers:

76.98

72.72

75.71

74.27

Correct answer:

72.72

Explanation:

In order to find the area of the shaded region, you will need to first find the areas of the triangle and the rectangle.

Recall how to find the area of a rectangle.

$\text{Area}=\text{length}\times\text{width}$

Now, in order to find the length of the rectangle, we will first need to find the length of the hypotenuse of the triangle. The question provides us with the following relationship between the lengths of the rectangle and hypotenuse:

$\text{length}=\frac{\text{hypotenuse}}{2}$

Recall how to find the hypotenuse of a right triangle:

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

Simplify.

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

Find the square root of each side.

$\text{Hypotenuse}=\sqrt{2(\text{side})^2}$

$\text{Hypotenuse}=\text{side}\sqrt2$

Substitute in the value of the side of the triangle to find the length of the hypotenuse.

$\text{Hypotenuse}=20\sqrt2$

Now, find the length of the rectangle:

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

$\text{length}=10\sqrt2$

Next, find the area of the rectangle:

$\text{Area of Rectangle}=10\sqrt2 \times 9$

$\text{Area of Rectangle}=90\sqrt2$

Now, recall how to find the area of a triangle:

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Since we are given a right isosceles triangle, the base and the height are the same length. Substitute in the given side length to find the area of the triangle.

$\text{Area of Triangle}=\frac{20\times 20}{2}$

$\text{Area of Triangle}=200$

Now that we have the areas of the rectangle and the triangle, we can find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Triangle}-\text{Area of Rectangle}$

$\text{Area of Shaded Region}=200-90\sqrt2$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=72.72$

Report an Error

Example Question #52 : Triangles

A rectangle is inscribed in a right isosceles triangle. The length of the rectangle is half the hypotenuse of the triangle and is shown in the provided figure. What is the area of the shaded region?

Possible Answers:

58.58

62.62

51.51

55.55

Correct answer:

58.58

Explanation:

In order to find the area of the shaded region, you will need to first find the areas of the triangle and the rectangle.

Recall how to find the area of a rectangle.

$\text{Area}=\text{length}\times\text{width}$

$\text{length}=\frac{\text{hypotenuse}}{2}$

Recall how to find the hypotenuse of a right triangle:

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

Simplify.

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

Find the square root of each side.

$\text{Hypotenuse}=\sqrt{2(\text{side})^2}$

$\text{Hypotenuse}=\text{side}\sqrt2$

Substitute in the value of the side of the triangle to find the length of the hypotenuse.

$\text{Hypotenuse}=20\sqrt2$

Now, find the length of the rectangle:

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

$\text{length}=10\sqrt2$

Next, find the area of the rectangle:

$\text{Area of Rectangle}=10\sqrt2 \times 10$

$\text{Area of Rectangle}=100\sqrt2$

Now, recall how to find the area of a triangle:

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Since we are given a right isosceles triangle, the base and the height are the same length. Substitute in the given side length to find the area of the triangle.

$\text{Area of Triangle}=\frac{20\times 20}{2}$

$\text{Area of Triangle}=200$

Now that we have the areas of the rectangle and the triangle, we can find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Triangle}-\text{Area of Rectangle}$

$\text{Area of Shaded Region}=200-100\sqrt2$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=58.58$

Report an Error

Example Question #51 : 45/45/90 Right Isosceles Triangles

A rectangle is inscribed in a right isosceles triangle. The length of the rectangle is half the hypotenuse of the triangle and is shown in the provided figure. What is the area of the shaded region?

Possible Answers:

179.35

184.58

169.21

154.29

Correct answer:

169.21

Explanation:

In order to find the area of the shaded region, you will need to first find the areas of the triangle and the rectangle.

Recall how to find the area of a rectangle.

$\text{Area}=\text{length}\times\text{width}$

$\text{length}=\frac{\text{hypotenuse}}{2}$

Recall how to find the hypotenuse of a right triangle:

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

Simplify.

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

Find the square root of each side.

$\text{Hypotenuse}=\sqrt{2(\text{side})^2}$

$\text{Hypotenuse}=\text{side}\sqrt2$

Substitute in the value of the side of the triangle to find the length of the hypotenuse.

$\text{Hypotenuse}=24\sqrt2$

Now, find the length of the rectangle:

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

$\text{length}=12\sqrt2$

Next, find the area of the rectangle:

$\text{Area of Rectangle}=12\sqrt2 \times 7$

$\text{Area of Rectangle}=84\sqrt2$

Now, recall how to find the area of a triangle:

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Since we are given a right isosceles triangle, the base and the height are the same length. Substitute in the given side length to find the area of the triangle.

$\text{Area of Triangle}=\frac{24\times 24}{2}$

$\text{Area of Triangle}=288$

Now that we have the areas of the rectangle and the triangle, we can find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Triangle}-\text{Area of Rectangle}$

$\text{Area of Shaded Region}=288-84\sqrt2$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=169.21$

Report an Error

Example Question #51 : Triangles

A rectangle is inscribed in a right isosceles triangle. The length of the rectangle is half the hypotenuse of the triangle and is shown in the provided figure. What is the area of the shaded region?

Possible Answers:

121.48

113.22

118.29

120.59

Correct answer:

118.29

Explanation:

In order to find the area of the shaded region, you will need to first find the areas of the triangle and the rectangle.

Recall how to find the area of a rectangle.

$\text{Area}=\text{length}\times\text{width}$

$\text{length}=\frac{\text{hypotenuse}}{2}$

Recall how to find the hypotenuse of a right triangle:

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

Simplify.

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

Find the square root of each side.

$\text{Hypotenuse}=\sqrt{2(\text{side})^2}$

$\text{Hypotenuse}=\text{side}\sqrt2$

Substitute in the value of the side of the triangle to find the length of the hypotenuse.

$\text{Hypotenuse}=24\sqrt2$

Now, find the length of the rectangle:

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

$\text{length}=12\sqrt2$

Next, find the area of the rectangle:

$\text{Area of Rectangle}=12\sqrt2 \times 10$

$\text{Area of Rectangle}=120\sqrt2$

Now, recall how to find the area of a triangle:

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Since we are given a right isosceles triangle, the base and the height are the same length. Substitute in the given side length to find the area of the triangle.

$\text{Area of Triangle}=\frac{24\times 24}{2}$

$\text{Area of Triangle}=288$

Now that we have the areas of the rectangle and the triangle, we can find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Triangle}-\text{Area of Rectangle}$

$\text{Area of Shaded Region}=288-120\sqrt2$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=118.29$

Report an Error

Example Question #55 : Triangles

A rectangle is inscribed in a right isosceles triangle. The length of the rectangle is half the hypotenuse of the triangle and is shown in the provided figure. What is the area of the shaded region?

Possible Answers:

150.86

151.29

143.26

152.24

Correct answer:

152.24

Explanation:

In order to find the area of the shaded region, you will need to first find the areas of the triangle and the rectangle.

Recall how to find the area of a rectangle.

$\text{Area}=\text{length}\times\text{width}$

$\text{length}=\frac{\text{hypotenuse}}{2}$

Recall how to find the hypotenuse of a right triangle:

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

Simplify.

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

Find the square root of each side.

$\text{Hypotenuse}=\sqrt{2(\text{side})^2}$

$\text{Hypotenuse}=\text{side}\sqrt2$

Substitute in the value of the side of the triangle to find the length of the hypotenuse.

$\text{Hypotenuse}=24\sqrt2$

Now, find the length of the rectangle:

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

$\text{length}=12\sqrt2$

Next, find the area of the rectangle:

$\text{Area of Rectangle}=12\sqrt2 \times 8$

$\text{Area of Rectangle}=96\sqrt2$

Now, recall how to find the area of a triangle:

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Since we are given a right isosceles triangle, the base and the height are the same length. Substitute in the given side length to find the area of the triangle.

$\text{Area of Triangle}=\frac{24\times 24}{2}$

$\text{Area of Triangle}=288$

Now that we have the areas of the rectangle and the triangle, we can find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Triangle}-\text{Area of Rectangle}$

$\text{Area of Shaded Region}=288-96\sqrt2$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=152.24$

Report an Error

Example Question #51 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem

A rectangle is inscribed in a right isosceles triangle. The length of the rectangle is half the hypotenuse of the triangle and is shown in the provided figure. What is the area of the shaded region?

Possible Answers:

90.58

79.41

80.61

88.23

Correct answer:

80.61

Explanation:

In order to find the area of the shaded region, you will need to first find the areas of the triangle and the rectangle.

Recall how to find the area of a rectangle.

$\text{Area}=\text{length}\times\text{width}$

$\text{length}=\frac{\text{hypotenuse}}{2}$

Recall how to find the hypotenuse of a right triangle:

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

Simplify.

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

Find the square root of each side.

$\text{Hypotenuse}=\sqrt{2(\text{side})^2}$

$\text{Hypotenuse}=\text{side}\sqrt2$

Substitute in the value of the side of the triangle to find the length of the hypotenuse.

$\text{Hypotenuse}=26\sqrt2$

Now, find the length of the rectangle:

$\text{length}=\frac{26\sqrt2}{2}$

$\text{length}=13\sqrt2$

Next, find the area of the rectangle:

$\text{Area of Rectangle}=13\sqrt2 \times 14$

$\text{Area of Rectangle}=182\sqrt2$

Now, recall how to find the area of a triangle:

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Since we are given a right isosceles triangle, the base and the height are the same length. Substitute in the given side length to find the area of the triangle.

$\text{Area of Triangle}=\frac{26\times 26}{2}=338$

Now that we have the areas of the rectangle and the triangle, we can find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Triangle}-\text{Area of Rectangle}$

$\text{Area of Shaded Region}=338-182\sqrt2$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=80.61$

Report an Error

Example Question #54 : 45/45/90 Right Isosceles Triangles

A rectangle is inscribed in a right isosceles triangle. The length of the rectangle is half the hypotenuse of the triangle and is shown in the provided figure. What is the area of the shaded region?

Possible Answers:

148.20

169.59

153.02

157.84

Correct answer:

153.02

Explanation:

In order to find the area of the shaded region, you will need to first find the areas of the triangle and the rectangle.

Recall how to find the area of a rectangle.

$\text{Area}=\text{length}\times\text{width}$

$\text{length}=\frac{\text{hypotenuse}}{2}$

Recall how to find the hypotenuse of a right triangle:

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

Simplify.

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

Find the square root of each side.

$\text{Hypotenuse}=\sqrt{2(\text{side})^2}$

$\text{Hypotenuse}=\text{side}\sqrt2$

Substitute in the value of the side of the triangle to find the length of the hypotenuse.

$\text{Hypotenuse}=30\sqrt2$

Now, find the length of the rectangle:

$\text{length}=\frac{30\sqrt2}{2}$

$\text{length}=15\sqrt2$

Next, find the area of the rectangle:

$\text{Area of Rectangle}=15\sqrt2 \times 14$

$\text{Area of Rectangle}=210\sqrt2$

Now, recall how to find the area of a triangle:

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Since we are given a right isosceles triangle, the base and the height are the same length. Substitute in the given side length to find the area of the triangle.

$\text{Area of Triangle}=\frac{30\times 30}{2}$

$\text{Area of Triangle}=450$

Now that we have the areas of the rectangle and the triangle, we can find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Triangle}-\text{Area of Rectangle}$

$\text{Area of Shaded Region}=450-210\sqrt2$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=153.02$

Report an Error

Example Question #55 : 45/45/90 Right Isosceles Triangles

Right triangle 7

Refer to the above figure.

True or false: $\overline{AC}$ has length $16\sqrt{2}$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is $90^{\circ }$ , so

$m \angle A+ m \angle C = 90^{\circ }$

Substituting $45^{\circ }$ for $m \angle C$ :

$m \angle A+ 45^{\circ } = 90^{\circ }$

$m \angle A= 45^{\circ }$

This makes $\bigtriangleup ABC$ a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, the length of hypotenuse $\overline{AC}$ is equal to that of leg $\overline{BC}$ multiplied by $\sqrt{2}$ . Therefore,

$AC = AB \cdot \sqrt{2} = 16\sqrt{2}$ .

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Basic Geometry : How to find the length of the hypotenuse of a 45/45/90 right isosceles triangle : Pythagorean Theorem

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #51 : 45/45/90 Right Isosceles Triangles

Example Question #52 : Triangles

Example Question #51 : 45/45/90 Right Isosceles Triangles

Example Question #51 : Triangles

Example Question #55 : Triangles

Example Question #51 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem

Example Question #54 : 45/45/90 Right Isosceles Triangles

Example Question #55 : 45/45/90 Right Isosceles Triangles

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