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Basic Geometry : 45/45/90 Right Isosceles Triangles

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Example Question #22 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

In the figure, a right isosceles triangle is placed in a rectangle. What is the area of the shaded region?

Possible Answers:

225

195

215

205

Correct answer:

195

Explanation:

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

First, recall how to find the area of a triangle.

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

In the figure, we are given the triangle’s hypotenuse, which is also the width of the rectangle. We can then use this information and the Pythagorean theorem to find the length of the sides of the triangle.

$\text{Hypotenuse}^2=\text{base}^2+\text{height}^2$

Now, because this is a right isosceles triangle,

$\text{base}=\text{height}$

We can then make the following substitution:

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

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

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

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

Therefore:

$\text{base}=\frac{\text{Hypotenuse}}{\sqrt2}$

$\text{base}=\frac{\text{Hypotenuse}\sqrt2}{2}$

Substitute in the value of the hypotenuse to find the length of the base:

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

$\text{base}=13\sqrt2$

Now, substitute this number in to find the area of the triangle.

$\text{Area of Triangle}=\frac{13\sqrt2 \times 13\sqrt2}{2}$

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

$\text{Area of Triangle}=169$

Next, recall how to find the area of the rectangle:

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

Substitute in the given information from the question to find the area.

$\text{Area of Rectangle}=14\times 26$

$\text{Area of Rectangle}=364$

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

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

$\text{Area of Shaded Region}=364-169$

Solve.

$\text{Area of Shaded Region}=195$

Report an Error

Example Question #23 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

In the figure, a right isosceles triangle is placed in a rectangle. What is the area of the shaded region?

Possible Answers:

135

145

150

165

Correct answer:

135

Explanation:

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

First, recall how to find the area of a triangle.

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

$\text{Hypotenuse}^2=\text{base}^2+\text{height}^2$

Now, because this is a right isosceles triangle,

$\text{base}=\text{height}$

We can then make the following substitution:

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

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

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

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

Therefore:

$\text{base}=\frac{\text{Hypotenuse}}{\sqrt2}$

$\text{base}=\frac{\text{Hypotenuse}\sqrt2}{2}$

Substitute in the value of the hypotenuse to find the length of the base:

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

$\text{base}=15\sqrt2$

Now, substitute this number in to find the area of the triangle.

$\text{Area of Triangle}=\frac{15\sqrt2 \times 15\sqrt2}{2}$

$\text{Area of Triangle}=\frac{450}{2}$

$\text{Area of Triangle}=225$

Next, recall how to find the area of the rectangle:

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

Substitute in the given information from the question to find the area.

$\text{Area of Rectangle}=12\times 30$

$\text{Area of Rectangle}=360$

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

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

$\text{Area of Shaded Region}=360-225$

Solve.

$\text{Area of Shaded Region}=135$

Report an Error

Example Question #24 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

In the figure, a right isosceles triangle is placed in a rectangle. What is the area of the shaded region?

Possible Answers:

180

150

120

Correct answer:

120

Explanation:

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

First, recall how to find the area of a triangle.

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

$\text{Hypotenuse}^2=\text{base}^2+\text{height}^2$

Now, because this is a right isosceles triangle,

$\text{base}=\text{height}$

We can then make the following substitution:

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

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

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

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

Therefore:

$\text{base}=\frac{\text{Hypotenuse}}{\sqrt2}$

$\text{base}=\frac{\text{Hypotenuse}\sqrt2}{2}$

Substitute in the value of the hypotenuse to find the length of the base:

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

$\text{base}=12\sqrt2$

Now, substitute this number in to find the area of the triangle.

$\text{Area of Triangle}=\frac{12\sqrt2 \times 12\sqrt2}{2}$

$\text{Area of Triangle}=\frac{288}{2}$

$\text{Area of Triangle}=144$

Next, recall how to find the area of the rectangle:

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

Substitute in the given information from the question to find the area.

$\text{Area of Rectangle}=11\times 24$

$\text{Area of Rectangle}=264$

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

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

$\text{Area of Shaded Region}=264-144$

Solve.

$\text{Area of Shaded Region}=120$

Report an Error

Example Question #31 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

If the length of one leg of a right isosceles triangle is 16cm, what is the length of the other leg?

Possible Answers:

$45\text{ cm}$

$32\text{ cm}$

$6\text{ cm}$

$8\text{ cm}$

$16\text{ cm}$

Correct answer:

$16\text{ cm}$

Explanation:

First we need to know that a right isosceles triangle has two legs with the same length.

In this case we do not need to worry about the hypotenuse.

Since one leg measures 16 cm, the other leg is the same length and so our answer is 16cm.

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Basic Geometry : 45/45/90 Right Isosceles Triangles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #22 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

Example Question #23 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

Example Question #24 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

Example Question #31 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

All Basic Geometry Resources

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