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

If the width of the rectangle is half the length of the hypotenuse of the triangle, then what is the area of the shaded region?

Explanation

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

First, let's recall how to find the area of a rectangle.

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

Now, the question tells us the following relationship between the width of the rectangle and the hypotenuse of the triangle:

$\text{Width}=\frac{\text{Hypotenuse}}{2}$

Now, let's use the Pythagorean theorem to find the length of the hypotenuse.

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

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

Substitute in the values of the base and of the height to find the hypotenuse.

$\text{Hypotenuse}=\sqrt{5^2+5^2}$

$\text{Hypotenuse}=5\sqrt2$

Now, substitute this value in to find the width of the rectangle.

$\text{Width}=\frac{5\sqrt2}{2}$

Now, find the area of the rectangle.

$\text{Area of Rectangle}=2\times\frac{5\sqrt2}{2}$

$\text{Area of Rectangle}=5\sqrt2$

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

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

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{5\times 5}{2}$

$\text{Area of Triangle}=\frac{25}{2}$

Finally, we are ready to find the area of the shaded region.

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

$\text{Area of Shaded Region}=\frac{25}{2}-5\sqrt2$

Solve.

$\text{Area of Shaded Region}=5.43$

If the width of the rectangle is half the hypotenuse of the triangle, then what is the area of the shaded region?

Explanation

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

First, let's recall how to find the area of a rectangle.

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

Now, the question tells us the following relationship between the width of the rectangle and the hypotenuse of the triangle:

$\text{Width}=\frac{\text{Hypotenuse}}{2}$

Now, let's use the Pythagorean theorem to find the length of the hypotenuse.

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

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

Substitute in the values of the base and of the height to find the hypotenuse.

$\text{Hypotenuse}=\sqrt{9^2+9^2}$

$\text{Hypotenuse}=9\sqrt2$

Now, substitute this value in to find the width of the rectangle.

$\text{Width}=\frac{9\sqrt2}{2}$

Now, find the area of the rectangle.

$\text{Area of Rectangle}=5\times\frac{9\sqrt2}{2}$

$\text{Area of Rectangle}=\frac{45\sqrt2}{2}$

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

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

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{9\times 9}{2}$

$\text{Area of Triangle}=\frac{81}{2}$

Finally, we are ready to find the area of the shaded region.

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

$\text{Area of Shaded Region}=\frac{81}{2}-\frac{45\sqrt2}{2}$

Solve.

$\text{Area of Shaded Region}=8.68$

Find the length of the hypotenuse.

$12\sqrt2$

$2\sqrt6$

$4\sqrt6$

$12\sqrt3$

Explanation

To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of and and a hypotenuse of ,

$c=\sqrt{a^2+b^2}$

Now, plug in the values of and from the question.

$\text{Hypotenuse}=\sqrt{12^2+12^2}$

$\text{Hypotenuse}=\sqrt{144+144}$

Solve.

$\text{Hypotenuse}{}=\sqrt{288}$

Simplify.

$\text{Hypotenuse}=12\sqrt2$

Find the length of the hypotenuse.

$13\sqrt2$

$13\sqrt3$

$\sqrt{26}$

Explanation

To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of and and a hypotenuse of ,

$c=\sqrt{a^2+b^2}$

Now, plug in the values of and from the question.

$\text{Hypotenuse}=\sqrt{13^2+13^2}$

$\text{Hypotenuse}=\sqrt{169+169}$

Solve.

$\text{Hypotenuse}=\sqrt{338}$

Simplify.

$\text{Hypotenuse}=13\sqrt2$

A right isosceles triangle has leg lengths of $\frac{1}{8}$ . What is the length of the hypotenuse?

$\frac{\sqrt2}{8}$

$8\sqrt2$

$4\sqrt2$

$12\sqrt2$

Explanation

Recall the Pythagorean Theorem:

Since we know that this is an isosceles right triangle, we know the following:

The Pythagorean Theorem can then be simplifed to the following equation:

Now, solve for since the question asks for the length of the hypotenuse.

$c=\sqrt{2a^2}$

$c=a\sqrt2$

Now, plug in the given value for to find the length of the hypotenuse.

$c=\frac{\sqrt2}{8}$

If the width of the rectangle is half the hypotenuse of the triangle, then what is the area of the shaded region?

Explanation

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

First, let's recall how to find the area of a rectangle.

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

Now, the question tells us the following relationship between the width of the rectangle and the hypotenuse of the triangle:

$\text{Width}=\frac{\text{Hypotenuse}}{2}$

Now, let's use the Pythagorean theorem to find the length of the hypotenuse.

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

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

Substitute in the values of the base and of the height to find the hypotenuse.

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

$\text{Hypotenuse}=2\sqrt2$

Now, substitute this value in to find the width of the rectangle.

$\text{Width}=\frac{2\sqrt2}{2}$

$\text{Width}=\sqrt2$

Now, find the area of the rectangle.

$\text{Area of Rectangle}=1\times\sqrt2$

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

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

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

Substitute in the given base and height to find the area.

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

$\text{Area of Triangle}=2$

Finally, we are ready to find the area of the shaded region.

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

$\text{Area of Shaded Region}=2-\sqrt2$

Solve.

$\text{Area of Shaded Region}=0.59$

Find the length of the hypotenuse.

$19\sqrt2$

$19\sqrt3$

$28\sqrt2$

Explanation

To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of and and a hypotenuse of ,

$c=\sqrt{a^2+b^2}$

Now, plug in the values of and from the question.

$\text{Hypotenuse}=\sqrt{19^2+19^2}$

$\text{Hypotenuse}=\sqrt{361+361}$

Solve.

$\text{Hypotenuse}=\sqrt{722}$

Simplify.

$\text{Hypotenuse}=19\sqrt2$

A right isosceles triangle has leg lengths of $2\sqrt{2}$ . What is the length of the hypotenuse?

$4\sqrt2$

Explanation

Recall the Pythagorean Theorem:

Since we know that this is an isosceles right triangle, we know the following:

The Pythagorean Theorem can then be simplifed to the following equation:

Now, solve for since the question asks for the length of the hypotenuse.

$c=\sqrt{2a^2}$

$c=a\sqrt2$

Now, plug in the given value for to find the length of the hypotenuse.

$c=(2\sqrt2)(\sqrt2)$

Simplify.

A right isosceles triangle has leg lengths of . In terms of , what is the length of the hypotenuse?

$x\sqrt2$

$\sqrt{2x}$

$\frac{x\sqrt2}{2}$

Explanation

Recall the Pythagorean Theorem:

Since we know that this is an isosceles right triangle, we know the following:

The Pythagorean Theorem can then be simplifed to the following equation:

Now, solve for since the question asks for the length of the hypotenuse.

$c=\sqrt{2a^2}$

$c=a\sqrt2$

Now, plug in the given value for to find the length of the hypotenuse.

$c=x\sqrt2$

A right isosceles triangle has leg lengths of $\frac{3}{4}$ . What is the length of the hypotenuse?

$\frac{3\sqrt2}{4}$

$\frac{2\sqrt2}{3}$

$\frac{4\sqrt3}{3}$

$\frac{\sqrt2}{6}$

Explanation

Recall the Pythagorean Theorem:

Since we know that this is an isosceles right triangle, we know the following:

The Pythagorean Theorem can then be simplifed to the following equation:

Now, solve for since the question asks for the length of the hypotenuse.

$c=\sqrt{2a^2}$

$c=a\sqrt2$

Now, plug in the given value for to find the length of the hypotenuse.

$c=\frac{3\sqrt2}{4}$

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

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