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Basic Geometry : How to find the height of a 45/45/90 right isosceles triangle

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Example Question #63 : Triangles

If the hypotenuse of an isoceles right triangle is , what is the length of the height?

Possible Answers:

$44\sqrt2$

$22\sqrt2$

$66\sqrt2$

Correct answer:

$44\sqrt2$

Explanation:

An isoceles right triangle is another way of saying that the triangle is a 45-45-90 triangle.

Now, recall the Pythagorean theorem:

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

Because we are working with a 45-45-90 triangle, the base and the height have the same length. We can rewrite the above equation as the following:

$\text{height}^2+\text{height}^2=\text{hypotenuse}^2$

$2(\text{height})^2=\text{hypotenuse}^2$

$\text{height}^2=\frac{\text{hypotenuse}^2}{2}$

$\text{height}=\sqrt{\frac{\text{hypotenuse}^2}{2}}$

Simplify.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}$

Multiply the fraction by one in the form of:

$1\rightarrow \frac{\sqrt{2}}{\sqrt{2}}$

Substitute.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\text{height}=\frac{\text{hypotenuse}(\sqrt2)}{2}{}$

Now, substitute in the value of the hypotenuse to find the height for the given triangle.

$\text{height}=\frac{88\sqrt2}{2}$

Simplify.

$\text{height}=44\sqrt2$

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Example Question #64 : Triangles

If the hypotenuse of an isoceles right triangle is 124 , what is the length of the height?

Possible Answers:

124

$62\sqrt2$

$62\sqrt3$

Correct answer:

$62\sqrt2$

Explanation:

An isoceles right triangle is another way of saying that the triangle is a 45-45-90 triangle.

Now, recall the Pythagorean theorem:

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

Because we are working with a 45-45-90 triangle, the base and the height have the same length. We can rewrite the above equation as the following:

$\text{height}^2+\text{height}^2=\text{hypotenuse}^2$

$2(\text{height})^2=\text{hypotenuse}^2$

$\text{height}^2=\frac{\text{hypotenuse}^2}{2}$

$\text{height}=\sqrt{\frac{\text{hypotenuse}^2}{2}}$

Simplify.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}$

Multiply the fraction by one in the form of:

$1\rightarrow \frac{\sqrt{2}}{\sqrt{2}}$

Substitute.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\text{height}=\frac{\text{hypotenuse}(\sqrt2)}{2}{}$

Now, substitute in the value of the hypotenuse to find the height for the given triangle.

$\text{height}=\frac{124\sqrt2}{2}$

Simplify.

$\text{height}=62\sqrt2$

Report an Error

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

If the hypotenuse of an isoceles right triangle is 144 , what is the length of the height?

Possible Answers:

$72\sqrt2$

$24\sqrt3$

$88\sqrt2$

Correct answer:

$72\sqrt2$

Explanation:

An isoceles right triangle is another way of saying that the triangle is a 45-45-90 triangle.

Now, recall the Pythagorean theorem:

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

Because we are working with a 45-45-90 triangle, the base and the height have the same length. We can rewrite the above equation as the following:

$\text{height}^2+\text{height}^2=\text{hypotenuse}^2$

$2(\text{height})^2=\text{hypotenuse}^2$

$\text{height}^2=\frac{\text{hypotenuse}^2}{2}$

$\text{height}=\sqrt{\frac{\text{hypotenuse}^2}{2}}$

Simplify.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}$

Multiply the fraction by one in the form of:

$1\rightarrow \frac{\sqrt{2}}{\sqrt{2}}$

Substitute.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\text{height}=\frac{\text{hypotenuse}(\sqrt2)}{2}{}$

Now, substitute in the value of the hypotenuse to find the height for the given triangle.

$\text{height}=\frac{144\sqrt2}{2}$

Simplify.

$\text{height}=72\sqrt2$

Report an Error

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

If the hypotenuse of an isoceles right triangle is , what is the length of the height of the triangle?

Possible Answers:

$95\sqrt2$

$\frac{95\sqrt2}{2}$

$95\sqrt3$

Correct answer:

$\frac{95\sqrt2}{2}$

Explanation:

An isoceles right triangle is another way of saying that the triangle is a 45-45-90 triangle.

Now, recall the Pythagorean theorem:

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

Because we are working with a 45-45-90 triangle, the base and the height have the same length. We can rewrite the above equation as the following:

$\text{height}^2+\text{height}^2=\text{hypotenuse}^2$

$2(\text{height})^2=\text{hypotenuse}^2$

$\text{height}^2=\frac{\text{hypotenuse}^2}{2}$

$\text{height}=\sqrt{\frac{\text{hypotenuse}^2}{2}}$

Simplify.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}$

Multiply the fraction by one in the form of:

$1\rightarrow \frac{\sqrt{2}}{\sqrt{2}}$

Substitute.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\text{height}=\frac{\text{hypotenuse}(\sqrt2)}{2}{}$

Now, substitute in the value of the hypotenuse to find the height for the given triangle.

$\text{height}=\frac{95\sqrt2}{2}$

Report an Error

Example Question #73 : 45/45/90 Right Isosceles Triangles

If the hypotenuse of an isoceles right triangle is , what is the length of the height?

Possible Answers:

$13\sqrt3$

$26\sqrt2$

$16\sqrt3$

Correct answer:

$26\sqrt2$

Explanation:

An isoceles right triangle is another way of saying that the triangle is a 45-45-90 triangle.

Now, recall the Pythagorean theorem:

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

Because we are working with a 45-45-90 triangle, the base and the height have the same length. We can rewrite the above equation as the following:

$\text{height}^2+\text{height}^2=\text{hypotenuse}^2$

$2(\text{height})^2=\text{hypotenuse}^2$

$\text{height}^2=\frac{\text{hypotenuse}^2}{2}$

$\text{height}=\sqrt{\frac{\text{hypotenuse}^2}{2}}$

Simplify.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}$

Multiply the fraction by one in the form of:

$1\rightarrow \frac{\sqrt{2}}{\sqrt{2}}$

Substitute.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\text{height}=\frac{\text{hypotenuse}(\sqrt2)}{2}{}$

Now, substitute in the value of the hypotenuse to find the height for the given triangle.

$\text{height}=\frac{52\sqrt2}{2}$

Simplify.

$\text{height}=26\sqrt2$

Report an Error

Example Question #13 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

Find the height of this triangle:

Isosceles right

Possible Answers:

12.25

4.77

1.87

Correct answer:

4.77

Explanation:

To find the height, use the Pythagorean Theorem. One of the legs is the missing side, and the other is 1.5, half of 3. The hypotenuse is 5:

(1.5)^2 + x^2 = 5^ 2

2.25 + x^2 = 25

x^2 = 22.75

$x \approx 4.77$

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Basic Geometry : How to find the height of a 45/45/90 right isosceles triangle

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #63 : Triangles

Example Question #64 : Triangles

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

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

Example Question #73 : 45/45/90 Right Isosceles Triangles

Example Question #13 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

All Basic Geometry Resources

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