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Basic Geometry : 45/45/90 Right Isosceles Triangles

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Example Questions

Example Question #171 : 45/45/90 Right Isosceles Triangles

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

Possible Answers:

$\displaystyle 16$

$2\sqrt3$ $\displaystyle 2\sqrt3$

$\sqrt3$ $\displaystyle \sqrt3$

$\displaystyle 12$

Correct answer:

$\sqrt3$ $\displaystyle \sqrt3$

Explanation:

A right isosceles triangle is also a $\displaystyle 45-45-90$ triangle.

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

$\displaystyle a^2+b^2=c^2$

Since this is an isosceles triangle,

a=b $\displaystyle a=b$

The Pythagorean Theorem can then be rewritten as the following:

$\displaystyle a^2+a^2=c^2$

$\displaystyle 2a^2=c^2$

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

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

Simplify.

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

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

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

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

Solve.

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

Now, substitute in the length of the hypotenuse in for $\displaystyle c$ to solve for the side of the triangle in the question.

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

Simplify.

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

Reduce.

$a=\sqrt3$ $\displaystyle a=\sqrt3$

Report an Error

Example Question #7 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

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

Possible Answers:

$\displaystyle 2$

$\sqrt3$ $\displaystyle \sqrt3$

$\displaystyle 4$

$\sqrt2$ $\displaystyle \sqrt2$

Correct answer:

$\displaystyle 2$

Explanation:

A right isosceles triangle is also a $\displaystyle 45-45-90$ triangle.

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

$\displaystyle a^2+b^2=c^2$

Since this is an isosceles triangle,

a=b $\displaystyle a=b$

The Pythagorean Theorem can then be rewritten as the following:

$\displaystyle a^2+a^2=c^2$

$\displaystyle 2a^2=c^2$

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

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

Simplify.

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

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

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

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

Solve.

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

Now, substitute in the length of the hypotenuse in for $\displaystyle c$ to solve for the side of the triangle in the question.

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

Simplify.

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

Reduce.

a=2 $\displaystyle a=2$

Report an Error

Example Question #4 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

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

Possible Answers:

$\sqrt5$ $\displaystyle \sqrt5$

$\displaystyle 2$

$\sqrt3$ $\displaystyle \sqrt3$

$\sqrt6$ $\displaystyle \sqrt6$

Correct answer:

$\sqrt5$ $\displaystyle \sqrt5$

Explanation:

A right isosceles triangle is also a $\displaystyle 45-45-90$ triangle.

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

$\displaystyle a^2+b^2=c^2$

Since this is an isosceles triangle,

a=b $\displaystyle a=b$

The Pythagorean Theorem can then be rewritten as the following:

$\displaystyle a^2+a^2=c^2$

$\displaystyle 2a^2=c^2$

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

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

Simplify.

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

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

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

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

Solve.

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

Now, substitute in the length of the hypotenuse in for $\displaystyle c$ to solve for the side of the triangle in the question.

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

Simplify.

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

Reduce.

$a=\sqrt5$ $\displaystyle a=\sqrt5$

Report an Error

Example Question #11 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

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

Possible Answers:

$\sqrt7$ $\displaystyle \sqrt7$

$2\sqrt{7}$ $\displaystyle 2\sqrt{7}$

$\sqrt{15}$ $\displaystyle \sqrt{15}$

$2\sqrt2$ $\displaystyle 2\sqrt2$

Correct answer:

$\sqrt7$ $\displaystyle \sqrt7$

Explanation:

A right isosceles triangle is also a $\displaystyle 45-45-90$ triangle.

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

$\displaystyle a^2+b^2=c^2$

Since this is an isosceles triangle,

a=b $\displaystyle a=b$

The Pythagorean Theorem can then be rewritten as the following:

$\displaystyle a^2+a^2=c^2$

$\displaystyle 2a^2=c^2$

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

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

Simplify.

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

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

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

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

Solve.

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

Now, substitute in the length of the hypotenuse in for $\displaystyle c$ to solve for the side of the triangle in the question.

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

Simplify.

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

Reduce.

$a=\sqrt7$ $\displaystyle a=\sqrt7$

Report an Error

Example Question #172 : Triangles

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

Possible Answers:

$\displaystyle 18$

$\displaystyle 6$

$\displaystyle 12$

$\displaystyle 24$

Correct answer:

$\displaystyle 12$

Explanation:

A right isosceles triangle is also a $\displaystyle 45-45-90$ triangle.

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

$\displaystyle a^2+b^2=c^2$

Since this is an isosceles triangle,

a=b $\displaystyle a=b$

The Pythagorean Theorem can then be rewritten as the following:

$\displaystyle a^2+a^2=c^2$

$\displaystyle 2a^2=c^2$

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

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

Simplify.

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

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

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

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

Solve.

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

Now, substitute in the length of the hypotenuse in for $\displaystyle c$ to solve for the side of the triangle in the question.

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

Simplify.

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

Reduce.

a=12 $\displaystyle a=12$

Report an Error

Example Question #173 : Triangles

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

Possible Answers:

$\displaystyle 3$

$4\sqrt2$ $\displaystyle 4\sqrt2$

$\sqrt{10}$ $\displaystyle \sqrt{10}$

$2\sqrt3$ $\displaystyle 2\sqrt3$

Correct answer:

$\sqrt{10}$ $\displaystyle \sqrt{10}$

Explanation:

A right isosceles triangle is also a $\displaystyle 45-45-90$ triangle.

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

$\displaystyle a^2+b^2=c^2$

Since this is an isosceles triangle,

a=b $\displaystyle a=b$

The Pythagorean Theorem can then be rewritten as the following:

$\displaystyle a^2+a^2=c^2$

$\displaystyle 2a^2=c^2$

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

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

Simplify.

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

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

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

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

Solve.

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

Now, substitute in the length of the hypotenuse in for $\displaystyle c$ to solve for the side of the triangle in the question.

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

Simplify.

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

Reduce.

$a=\sqrt{10}$ $\displaystyle a=\sqrt{10}$

Report an Error

Example Question #12 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

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

Possible Answers:

$\displaystyle 12$

$8\sqrt{10}$ $\displaystyle 8\sqrt{10}$

$6\sqrt{14}$ $\displaystyle 6\sqrt{14}$

$3\sqrt{22}$ $\displaystyle 3\sqrt{22}$

Correct answer:

$6\sqrt{14}$ $\displaystyle 6\sqrt{14}$

Explanation:

A right isosceles triangle is also a $\displaystyle 45-45-90$ triangle.

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

$\displaystyle a^2+b^2=c^2$

Since this is an isosceles triangle,

a=b $\displaystyle a=b$

The Pythagorean Theorem can then be rewritten as the following:

$\displaystyle a^2+a^2=c^2$

$\displaystyle 2a^2=c^2$

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

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

Simplify.

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

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

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

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

Solve.

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

Now, substitute in the length of the hypotenuse in for $\displaystyle c$ to solve for the side of the triangle in the question.

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

Simplify.

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

Reduce.

$a=6\sqrt{14}$ $\displaystyle a=6\sqrt{14}$

Report an Error

Example Question #175 : Triangles

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

Possible Answers:

$2\sqrt6$ $\displaystyle 2\sqrt6$

$8\sqrt3$ $\displaystyle 8\sqrt3$

$6\sqrt2$ $\displaystyle 6\sqrt2$

$4\sqrt6$ $\displaystyle 4\sqrt6$

Correct answer:

$2\sqrt6$ $\displaystyle 2\sqrt6$

Explanation:

A right isosceles triangle is also a $\displaystyle 45-45-90$ triangle.

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

$\displaystyle a^2+b^2=c^2$

Since this is an isosceles triangle,

a=b $\displaystyle a=b$

The Pythagorean Theorem can then be rewritten as the following:

$\displaystyle a^2+a^2=c^2$

$\displaystyle 2a^2=c^2$

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

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

Simplify.

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

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

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

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

Solve.

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

Now, substitute in the length of the hypotenuse in for $\displaystyle c$ to solve for the side of the triangle in the question.

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

Simplify.

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

Reduce.

$a=2\sqrt6$ $\displaystyle a=2\sqrt6$

Report an Error

Example Question #176 : Triangles

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

Possible Answers:

$9\sqrt2$ $\displaystyle 9\sqrt2$

$9\sqrt3$ $\displaystyle 9\sqrt3$

$\frac{9\sqrt2}{2}$ $\displaystyle \frac{9\sqrt2}{2}$

$\frac{9\sqrt3}{2}$ $\displaystyle \frac{9\sqrt3}{2}$

Correct answer:

$9\sqrt3$ $\displaystyle 9\sqrt3$

Explanation:

A right isosceles triangle is also a $\displaystyle 45-45-90$ triangle.

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

$\displaystyle a^2+b^2=c^2$

Since this is an isosceles triangle,

a=b $\displaystyle a=b$

The Pythagorean Theorem can then be rewritten as the following:

$\displaystyle a^2+a^2=c^2$

$\displaystyle 2a^2=c^2$

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

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

Simplify.

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

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

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

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

Solve.

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

Now, substitute in the length of the hypotenuse in for $\displaystyle c$ to solve for the side of the triangle in the question.

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

Simplify.

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

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

Reduce.

$a=9\sqrt3$ $\displaystyle a=9\sqrt3$

Report an Error

Example Question #177 : Triangles

If the hypotenuse of a right isosceles triangle is 128 $\displaystyle 128$, what is the length of a side of the triangle?

Possible Answers:

$\displaystyle 64$

$64\sqrt2$ $\displaystyle 64\sqrt2$

128 $\displaystyle 128$

$32\sqrt3$ $\displaystyle 32\sqrt3$

Correct answer:

$64\sqrt2$ $\displaystyle 64\sqrt2$

Explanation:

A right isosceles triangle is also a $\displaystyle 45-45-90$ triangle.

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

$\displaystyle a^2+b^2=c^2$

Since this is an isosceles triangle,

a=b $\displaystyle a=b$

The Pythagorean Theorem can then be rewritten as the following:

$\displaystyle a^2+a^2=c^2$

$\displaystyle 2a^2=c^2$

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

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

Simplify.

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

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

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

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

Solve.

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

Now, substitute in the length of the hypotenuse in for $\displaystyle c$ to solve for the side of the triangle in the question.

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

Simplify.

$a=64\sqrt2$ $\displaystyle a=64\sqrt2$

Report an Error

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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 #171 : 45/45/90 Right Isosceles Triangles

Example Question #7 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

Example Question #4 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

Example Question #11 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

Example Question #172 : Triangles

Example Question #173 : Triangles

Example Question #12 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

Example Question #175 : Triangles

Example Question #176 : Triangles

Example Question #177 : Triangles

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