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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 #171 : Circles

Find the length of the radius of a circle given the circumference is $6\pi$ .

Possible Answers:

$12\pi$

$6\pi$

$3\pi$

Correct answer:

Explanation:

To solve, simply use the formula for the circumference of a circle and solve for r. Thus,

$C=2\pi{r}\Rightarrow r=\frac{C}{2\pi}$

$r=\frac{6\pi}{2\pi}=3$

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Example Question #172 : Circles

Find the radius of a circle inscribed in a square that has a diagonal of $12\sqrt2$ .

Possible Answers:

$6\sqrt2$

$3\sqrt2$

Correct answer:

Explanation:

Notice that the diagonal of the square is also the hypotenuse of a right isosceles triangle whose legs are also the sides of the square. You should also notice that the diameter of the circle has the same length as that of a side of the square.

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

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

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

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

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

Now, substitute in the value of the diagonal to find the length of a side of the square.

$\text{side}=\frac{12\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=12$

Now keep in mind the following relationship between the diameter and the side of the square:

$\text{Diameter}=\text{side}=12$

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(12)$

Solve.

$\text{Radius}=6$

Report an Error

Example Question #14 : How To Find The Length Of A Radius

Find the radius of a circle inscribed in a square that has a diagonal of $3\sqrt2$ .

Possible Answers:

$4\sqrt2$

$\frac{3\sqrt2}{2}$

$6\sqrt2$

$\frac{3}{2}$

Correct answer:

$\frac{3}{2}$

Explanation:

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

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

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

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

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

Now, substitute in the value of the diagonal to find the length of a side of the square.

$\text{side}=\frac{3\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=3$

Now keep in mind the following relationship between the diameter and the side of the square:

$\text{Diameter}=\text{side}=3$

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(3)$

Solve.

$\text{Radius}=\frac{3}{2}$

Report an Error

Example Question #173 : Circles

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

Possible Answers:

$\sqrt2$

Correct answer:

Explanation:

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

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

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

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

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

Now, substitute in the value of the diagonal to find the length of a side of the square.

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

Simplify.

$\text{side}=2$

Now keep in mind the following relationship between the diameter and the side of the square:

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

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(2)$

Solve.

$\text{Radius}=1$

Report an Error

Example Question #15 : How To Find The Length Of A Radius

Find the radius of a circle inscribed in a square with a diagonal of $6\sqrt2$ .

Possible Answers:

$6\sqrt2$

Correct answer:

Explanation:

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

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

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

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

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

Now, substitute in the value of the diagonal to find the length of a side of the square.

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

Simplify.

$\text{side}=6$

Now keep in mind the following relationship between the diameter and the side of the square:

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

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(6)$

Solve.

$\text{Radius}=3$

Report an Error

Example Question #17 : How To Find The Length Of A Radius

Find the length of the radius of a circle inscribed in a square with a diagonal of $5\sqrt2$ .

Possible Answers:

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

$3\sqrt{2}$

$\frac{5}{2}$

Correct answer:

$\frac{5}{2}$

Explanation:

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

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

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

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

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

Now, substitute in the value of the diagonal to find the length of a side of the square.

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

Simplify.

$\text{side}=5$

Now keep in mind the following relationship between the diameter and the side of the square:

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

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(5)$

Solve.

$\text{Radius}=\frac{5}{2}$

Report an Error

Example Question #18 : How To Find The Length Of A Radius

Find the length of the radius of a circle inscribed in a square with a diagonal of $4\sqrt2$ .

Possible Answers:

$2\sqrt2$

Correct answer:

Explanation:

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

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

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

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

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

Now, substitute in the value of the diagonal to find the length of a side of the square.

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

Simplify.

$\text{side}=4$

Now keep in mind the following relationship between the diameter and the side of the square:

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

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(4)$

Solve.

$\text{Radius}=2$

Report an Error

Example Question #19 : How To Find The Length Of A Radius

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

Possible Answers:

$\frac{7}{2}$

$15\sqrt2$

$\frac{7\sqrt2}{2}$

Correct answer:

$\frac{7}{2}$

Explanation:

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

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

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

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

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

Now, substitute in the value of the diagonal to find the length of a side of the square.

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

Simplify.

$\text{side}=7$

Now keep in mind the following relationship between the diameter and the side of the square:

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

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(7)$

Solve.

$\text{Radius}=\frac{7}{2}$

Report an Error

Example Question #20 : How To Find The Length Of A Radius

Find the length of the radius of a circle inscribed in a square that has a diagonal of $8\sqrt2$ .

Possible Answers:

Correct answer:

Explanation:

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

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

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

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

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

Now, substitute in the value of the diagonal to find the length of a side of the square.

$\text{side}=\frac{8\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=8$

Now keep in mind the following relationship between the diameter and the side of the square:

$\text{Diameter}=\text{side}=8$

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(8)$

Solve.

$\text{Radius}=4$

Report an Error

Example Question #21 : How To Find The Length Of A Radius

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

Possible Answers:

$\frac{\sqrt2}{4}$

$\frac{\sqrt2}{2}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{2}$

Explanation:

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

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

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

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

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

Now, substitute in the value of the diagonal to find the length of a side of the square.

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

Simplify.

$\text{side}=1$

Now keep in mind the following relationship between the diameter and the side of the square:

$\text{Diameter}=\text{side}=1$

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(1)$

Solve.

$\text{Radius}=\frac{1}{2}$

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 #171 : Circles

Example Question #172 : Circles

Example Question #14 : How To Find The Length Of A Radius

Example Question #173 : Circles

Example Question #15 : How To Find The Length Of A Radius

Example Question #17 : How To Find The Length Of A Radius

Example Question #18 : How To Find The Length Of A Radius

Example Question #19 : How To Find The Length Of A Radius

Example Question #20 : How To Find The Length Of A Radius

Example Question #21 : How To Find The Length Of A Radius

All Basic Geometry Resources

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