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Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

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9 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : 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?

Possible Answers:

$68\sqrt2$

$34\sqrt2$

$14\sqrt3$

$60\sqrt3$

Correct answer:

$34\sqrt2$

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.

$\text{height}=\frac{68\sqrt2}{2}$

Simplify.

$\text{height}=34\sqrt2$

Report an Error

Example Question #61 : 45/45/90 Right Isosceles Triangles

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

Possible Answers:

$22\sqrt2$

$44\sqrt2$

$66\sqrt2$

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.

$\text{height}=44\sqrt2$

Report an Error

Example Question #63 : 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 #71 : Triangles

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

Possible Answers:

$24\sqrt3$

$88\sqrt2$

$72\sqrt2$

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 #11 : 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:

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

$95\sqrt2$

$95\sqrt3$

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 #1051 : Basic Geometry

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

Possible Answers:

$13\sqrt3$

$16\sqrt3$

$26\sqrt2$

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 #71 : 45/45/90 Right Isosceles Triangles

Find the height of this triangle:

Isosceles right

Possible Answers:

4.77

12.25

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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Example Question #1051 : Basic Geometry

$What\;is\;the\;area\;of\;\Delta ABC?$

Possible Answers:

4.5

6.5

Correct answer:

4.5

Explanation:

$Area\;\Delta=\frac{1}{2}(base)(height)$

$\frac{1}{2}(3)(3)=\frac{9}{2}=4.5$

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Example Question #1051 : Plane Geometry

Consider an isosceles triangle with a height of 24 and a base of 12. What is the area of this triangle?

Isoc._triangle

Possible Answers:

169

155

144

121

Correct answer:

144

Explanation:

The formula for the area of a trianlge is A = base * height * (1/2).

We're lucky here, because the question gives us all of the values we need. We simply need to plug them in:

A = base * height * (1/2) = 12 * 24 * (1/2) = 12 * 12 = 144

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Example Question #2 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

Calculate the area of an isosceles right triangle who's hypotenuse is $4\sqrt{2}$ inches.

Possible Answers:

$4 \text{ in.}^2$

$2 \text{ in.}^2$

$8 \text{ in.}^2$

$16 \text{ in.}^2$

Correct answer:

$8 \text{ in.}^2$

Explanation:

The formula for the area of a triangle, right or not, is one half the base times height.

In this case, they are both $4 \text{ in.}$ Therefore, the respective values are entered, yielding:

$\frac{1}{2}(4\text{ in.})(4\text{ in.})=8 \text{ in.}^2$

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All Basic Geometry Resources

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Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #2 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

Example Question #61 : 45/45/90 Right Isosceles Triangles

Example Question #63 : Triangles

Example Question #71 : Triangles

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

Example Question #1051 : Basic Geometry

Example Question #71 : 45/45/90 Right Isosceles Triangles

Example Question #1051 : Basic Geometry

Example Question #1051 : Plane Geometry

Example Question #2 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

All Basic Geometry Resources

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