How to find the length of the side of a 45/45/90 right isosceles triangle - Geometry

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Question

The perimeter of a 45-45-90 triangle is 100 inches. To the nearest tenth of an inch, what is the length of each leg?

Answer

Let be the length of a leg; then the hypotenuse is $c=a\sqrt{2}$ , and the perimeter is

$P = a + a + a \sqrt{2} = (2 + \sqrt{2} ) a = 100$

Therefore,

$a = \frac{100}{2 + \sqrt{2}} \approx 29.2$

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Question

Angle in the triangle shown below is 45 degrees. Side has a length of 10. What is the length of side ?

Answer

Since we know two of the three angles in this triangle, we can calculate the third, .

Therefore this is a 45/45/90 right triangle. Remember that 45/45/90 right triangles are have a leg:leg:hypotenuse ratio of 1:1: $\small \sqrt{2}$ .

We know the hypotenuse, $\small c$ , so we can quickly calculate the length of one of the legs, $\small a$ , by dividing by $\small \sqrt{2}$ :

$\small a=\frac{c}{\sqrt{2}}=\frac{10}{\sqrt{2}}$

To make this look like one of the answer choies, rationalize the denominator by muliplying the fraction by $\frac{\sqrt{2}}{\sqrt{2}}$ :

$\frac{10}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{10\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}=\frac{10\sqrt{2}}{2}=5\sqrt{2}$

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Question

$\fn_cm In; \Delta ABC, \overline{BC}=3 ;and ;\angle BAC=45^{\circ}. ;What;is;the;length;of;\overline{AB}?$

Answer

$\fn_cm \Delta ABC;is;an;isosceles;triangle. ;In; an ;isosceles;triangle, two;sides;and ;their ;opposite;angles ;are;equal.$

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Question

$\fn_cm What;is;the;length;of;\overline{DE} ;in; \Delta DEF?$

Answer

$\fn_cm \Delta DEF ;is;a;45^{\circ}-45^{\circ}-90^{\circ};isoscles;triangle, ;which;have;sides;in;a;ratio;of;1:1:\sqrt2.$

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Question

$\Delta ABC$ is a $45^\circ-45^\circ-90^\circ$ triangle.

$m\angle B=90^\circ$

$\overline{AC}=4$

What is the length of $\overline{AB}$ ?

Answer

We know that the sides of $45^{\circ}-45^{\circ}-90^{\circ}$ triangles are in the ratio of $1:1:\sqrt{2}$ , where the shorter sides lies opposite the $45^{\circ}$ angles, and the longer side is the hypotenuse and lies opposite the right angle. We are given that the hypotenuse is .

Divide the length of the hypotenuse by $\sqrt{2}$ to calculate the ratio of magnification.

$\frac{4}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{4\sqrt{2}}{2}={2}\sqrt{2}$

Multiply the length of the shorter sides by the ratio of magnification.

$1\cdot2\sqrt{2}=2\sqrt{2}$

So the length of $\overline{AB}$ (and $\overline{BC}$ ) is $2\sqrt{2}$ .

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Question

The following image is not to scale.

Find the length of one of the legs of the right triangle.

Answer

Because of the tick marks on both legs, we can determine that this right triangle is a 45/45/90 triangle. Because the length of both legs are the same, this means that the angle opposite of each leg is also the same.

45/45/90 triangles are special, just like 30/60/90 triangles. Solving for one of the leg lengths can be determined easily through remembering the following:

Using this and the 7ft, we can solve for "s" which will provide us with the leg length.

$7ft=s\sqrt{2}$

$\frac{7}{\sqrt{2}}=s$ while this is the correct answer, the options provided are represented as simplified radicals.

$\frac{7}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}={\color{blue} \frac{7\sqrt{2}}{2}}$

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Question

If the hypotenuse of a right isosceles triangle is $\sqrt6$ , what is the length of a side of the triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{(\sqrt6)(\sqrt2)}{2}$

Simplify.

$a=\frac{\sqrt{12}}{2}$

Reduce.

$a=\sqrt3$

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Question

If the hypotenuse of a right isosceles triangle is $2\sqrt2$ , what is the length of a side of this triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{(2\sqrt2)(\sqrt2)}{2}$

Simplify.

$a=\frac{\sqrt{4}}{2}$

Reduce.

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Question

If the hypotenuse of a right isosceles triangle is $\sqrt{10}$ , what is the length of a side of this triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{(\sqrt{10})(\sqrt2)}{2}$

Simplify.

$a=\frac{\sqrt{20}}{2}$

Reduce.

$a=\sqrt5$

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Question

If the hypotenuse of a right isosceles triangle is $2\sqrt3$ , what is the length of a side of the triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{(2\sqrt3)(\sqrt2)}{2}$

Simplify.

$a=\frac{2\sqrt{6}}{2}$

Reduce.

$a=\sqrt6$

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Question

If the hypotenuse of a right isosceles triangle is $\sqrt{14}$ , what is the length of a side of the triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{(\sqrt{14})(\sqrt2)}{2}$

Simplify.

$a=\frac{\sqrt{28}}{2}$

Reduce.

$a=\sqrt7$

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Question

If the hypotenuse of a right isosceles triangle is $12\sqrt2$ , what is the length of a side of the triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{(12\sqrt2)(\sqrt2)}{2}$

Simplify.

$a=\frac{24}{2}$

Reduce.

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Question

If the hypotenuse of a right isosceles triangle is $2\sqrt5$ , what is the length of a side of the triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{(2\sqrt5)(\sqrt2)}{2}$

Simplify.

$a=\frac{2\sqrt{10}}{2}$

Reduce.

$a=\sqrt{10}$

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Question

If the hypotenuse of a right isosceles triangle is $12\sqrt{7}$ , what is the length of one side of the triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{(12\sqrt7)(\sqrt2)}{2}$

Simplify.

$a=\frac{12\sqrt{14}}{2}$

Reduce.

$a=6\sqrt{14}$

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Question

If the hypotenuse of a right isosceles triangle is $4\sqrt{3}$ , what is the length of one side of the triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{(4\sqrt3)(\sqrt2)}{2}$

Simplify.

$a=\frac{4\sqrt{6}}{2}$

Reduce.

$a=2\sqrt6$

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Question

If the hypotenuse of a right isosceles triangle is $9\sqrt6$ , what is the length of one side of the triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{(9\sqrt6)(\sqrt2)}{2}$

Simplify.

$a=\frac{9\sqrt{12}}{2}$

$a=\frac{18\sqrt3}{2}$

Reduce.

$a=9\sqrt3$

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Question

If the hypotenuse of a right isosceles triangle is , what is the length of a side of the triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{128(\sqrt2)}{2}$

Simplify.

$a=64\sqrt2$

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Question

If the hypotenuse of a right isosceles triangle is , what is the length of a side of the triangle?

Answer

A right isosceles triangle is also a triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

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

Simplify.

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

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

Multiply the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a=\frac{c}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$a=\frac{c\sqrt2}{2}$

Now, substitute in the length of the hypotenuse in for to solve for the side of the triangle in the question.

$a=\frac{1000(\sqrt2)}{2}$

Simplify.

$a=500\sqrt2$

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Question

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

Answer

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:

$\text{base}=\frac{18\sqrt2}{2}$

$\text{base}=9\sqrt2$

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

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

$\text{Area of Triangle}=\frac{162}{2}$

$\text{Area of Triangle}=81$

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.

$\text{Area of Rectangle}=12\times18$

$\text{Area of Rectangle}=216$

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

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

$\text{Area of Shaded Region}=216-81$

Solve.

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

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Question

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

Answer

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:

$\text{base}=\frac{34\sqrt2}{2}$

$\text{base}=17\sqrt2$

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

$\text{Area of Triangle}=\frac{17\sqrt2 \times 17\sqrt2}{2}$

$\text{Area of Triangle}=\frac{578}{2}$

$\text{Area of Triangle}=289$

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.

$\text{Area of Rectangle}=15\times34$

$\text{Area of Rectangle}=510$

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

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

$\text{Area of Shaded Region}=510-289$

Solve.

$\text{Area of Shaded Region}=221$

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