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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 #341 : Circles

Find the length of the diameter of a circle inscribed in a square that has a diagonal of $2\sqrt2$ .

Possible Answers:

$2\pi$

$4\pi$

Correct answer:

Explanation:

When you draw out the circle that is inscribed in a square, you should notice two things. The first thing you should notice is that the diagonal of the square is also the hypotenuse of a right isosceles triangle that has the side lengths of the square as its legs. The second thing you should notice is that the diameter of the circle has the same length as the length of one side of the square.

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{2\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=2$

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Example Question #342 : Basic Geometry

Find the length of the diameter of a circle inscribed in a square that has a diagonal of $4\sqrt2$ .

Possible Answers:

$16\pi$

$2\pi$

Correct answer:

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{4\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=4$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=4$

Report an Error

Example Question #343 : Basic Geometry

Find the length of the diameter of a circle inscribed in a square that has a diagonal of $5\sqrt2$ .

Possible Answers:

$10\pi$

$5\sqrt2$

$\frac{5\sqrt2}{2}$

Correct answer:

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{5\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=5$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=5$

Report an Error

Example Question #344 : Basic Geometry

Find the length of the diameter of a circle inscribed in a square that has a diagonal of $6\sqrt2$ .

Possible Answers:

$3\pi$

$12\pi$

Correct answer:

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{6\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=6$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=6$

Report an Error

Example Question #345 : Basic Geometry

Find the length of the diameter of a circle inscribed in a square that has a diagonal of $7\sqrt2$ .

Possible Answers:

$\frac{7}{2}$

$7\sqrt2$

Correct answer:

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{7\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=7$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=7$

Report an Error

Example Question #41 : How To Find The Length Of The Diameter

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$5\sqrt2$

$25\pi$

$10\pi$

Correct answer:

$5\sqrt2$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{10(\sqrt2)}{2}$

Simplify.

$\text{side}=5\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=5\sqrt2$

Report an Error

Example Question #42 : Diameter

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$10\sqrt2$

200

$5\sqrt2$

Correct answer:

$10\sqrt2$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{20(\sqrt2)}{2}$

Simplify.

$\text{side}=10\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=10\sqrt2$

Report an Error

Example Question #43 : Diameter

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$6\sqrt2$

$18\pi$

$24\pi$

$12\sqrt2$

Correct answer:

$12\sqrt2$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{24(\sqrt2)}{2}$

Simplify.

$\text{side}=12\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=12\sqrt2$

Report an Error

Example Question #44 : Diameter

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$13\sqrt2$

$\frac{13}{2}\sqrt2$

$14\sqrt2$

$26\pi$

Correct answer:

$13\sqrt2$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{26(\sqrt2)}{2}$

Simplify.

$\text{side}=13\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=13\sqrt2$

Report an Error

Example Question #342 : Circles

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$54\sqrt2$

$18\sqrt2$

Correct answer:

$18\sqrt2$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{36(\sqrt2)}{2}$

Simplify.

$\text{side}=18\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=18\sqrt2$

Report an Error

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Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #341 : Circles

Example Question #342 : Basic Geometry

Example Question #343 : Basic Geometry

Example Question #344 : Basic Geometry

Example Question #345 : Basic Geometry

Example Question #41 : How To Find The Length Of The Diameter

Example Question #42 : Diameter

Example Question #43 : Diameter

Example Question #44 : Diameter

Example Question #342 : Circles

All Basic Geometry Resources

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