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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 #22 : How To Find The Length Of The Diagonal Of A Rectangle

If a rectangle is inscribed in a circle with a circumference of $\sqrt3\pi$ $\displaystyle \sqrt3\pi$, what is the length of the diagonal of the rectangle?

Possible Answers:

$\displaystyle 3$

$\sqrt3$ $\displaystyle \sqrt3$

$\displaystyle 6$

$\displaystyle 9$

Correct answer:

$\sqrt3$ $\displaystyle \sqrt3$

Explanation:

You should notice that the diagonal of the rectangle is the same as the diameter of the circle.

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\text{diameter}\times\pi$ $\displaystyle \text{Circumference}=\text{diameter}\times\pi$

Since we are given the circumference and need the diameter, rerwite the equation to solve for the diameter.

$\text{Diameter}=\frac{\text{Circumference}}{\pi}$ $\displaystyle \text{Diameter}=\frac{\text{Circumference}}{\pi}$

Plug in the given circumference to find the diameter.

$\text{Diameter}=\frac{\sqrt3\pi}{\pi}=\sqrt3$ $\displaystyle \text{Diameter}=\frac{\sqrt3\pi}{\pi}=\sqrt3$

Now, recall the relationship between the diameter of the circle and the diagonal of the rectangle:

$\text{Diameter}=\text{Diagonal}=\sqrt3$ $\displaystyle \text{Diameter}=\text{Diagonal}=\sqrt3$

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Example Question #23 : How To Find The Length Of The Diagonal Of A Rectangle

If a rectangle is inscribed in a circle with a circumference of $8\pi$ $\displaystyle 8\pi$, what is the length of the diagonal of the rectangle?

Possible Answers:

$\displaystyle 4$

$2\pi$ $\displaystyle 2\pi$

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

$\displaystyle 8$

Correct answer:

$\displaystyle 8$

Explanation:

You should notice that the diagonal of the rectangle is the same as the diameter of the circle.

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\text{diameter}\times\pi$ $\displaystyle \text{Circumference}=\text{diameter}\times\pi$

Since we are given the circumference and need the diameter, rerwite the equation to solve for the diameter.

$\text{Diameter}=\frac{\text{Circumference}}{\pi}$ $\displaystyle \text{Diameter}=\frac{\text{Circumference}}{\pi}$

Plug in the given circumference to find the diameter.

$\text{Diameter}=\frac{8\pi}{\pi}=8$ $\displaystyle \text{Diameter}=\frac{8\pi}{\pi}=8$

Now, recall the relationship between the diameter of the circle and the diagonal of the rectangle:

$\text{Diameter}=\text{Diagonal}=8$ $\displaystyle \text{Diameter}=\text{Diagonal}=8$

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Example Question #24 : How To Find The Length Of The Diagonal Of A Rectangle

If a rectangle is inscribed in a circle with a circumference of $10\pi$ $\displaystyle 10\pi$, what is the length of the diagonal of the rectangle?

Possible Answers:

$\displaystyle 10$

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

$100\pi$ $\displaystyle 100\pi$

$\displaystyle 5$

Correct answer:

$\displaystyle 10$

Explanation:

You should notice that the diagonal of the rectangle is the same as the diameter of the circle.

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\text{diameter}\times\pi$ $\displaystyle \text{Circumference}=\text{diameter}\times\pi$

Since we are given the circumference and need the diameter, rerwite the equation to solve for the diameter.

$\text{Diameter}=\frac{\text{Circumference}}{\pi}$ $\displaystyle \text{Diameter}=\frac{\text{Circumference}}{\pi}$

Plug in the given circumference to find the diameter.

$\text{Diameter}=\frac{10\pi}{\pi}=10$ $\displaystyle \text{Diameter}=\frac{10\pi}{\pi}=10$

Now, recall the relationship between the diameter of the circle and the diagonal of the rectangle:

$\text{Diameter}=\text{Diagonal}=10$ $\displaystyle \text{Diameter}=\text{Diagonal}=10$

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Example Question #24 : How To Find The Length Of The Diagonal Of A Rectangle

If a rectangle is inscribed in a circle with a circumference of $15\pi$ $\displaystyle 15\pi$, what is the length of the diagonal of the rectangle?

Possible Answers:

225 $\displaystyle 225$

$30\pi$ $\displaystyle 30\pi$

$\displaystyle 15$

$\frac{15\pi}{2}$ $\displaystyle \frac{15\pi}{2}$

Correct answer:

$\displaystyle 15$

Explanation:

You should notice that the diagonal of the rectangle is the same as the diameter of the circle.

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\text{diameter}\times\pi$ $\displaystyle \text{Circumference}=\text{diameter}\times\pi$

Since we are given the circumference and need the diameter, rerwite the equation to solve for the diameter.

$\text{Diameter}=\frac{\text{Circumference}}{\pi}$ $\displaystyle \text{Diameter}=\frac{\text{Circumference}}{\pi}$

Plug in the given circumference to find the diameter.

$\text{Diameter}=\frac{15\pi}{\pi}=15$ $\displaystyle \text{Diameter}=\frac{15\pi}{\pi}=15$

Now, recall the relationship between the diameter of the circle and the diagonal of the rectangle:

$\text{Diameter}=\text{Diagonal}=15$ $\displaystyle \text{Diameter}=\text{Diagonal}=15$

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Example Question #25 : How To Find The Length Of The Diagonal Of A Rectangle

If a rectangle is inscribed in a circle with a circumference of $100\pi$ $\displaystyle 100\pi$, what is the length of the diagonal of the rectangle?

Possible Answers:

$\displaystyle 10$

10000 $\displaystyle 10000$

$\displaystyle 50$

100 $\displaystyle 100$

Correct answer:

100 $\displaystyle 100$

Explanation:

You should notice that the diagonal of the rectangle is the same as the diameter of the circle.

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\text{diameter}\times\pi$ $\displaystyle \text{Circumference}=\text{diameter}\times\pi$

Since we are given the circumference and need the diameter, rerwite the equation to solve for the diameter.

$\text{Diameter}=\frac{\text{Circumference}}{\pi}$ $\displaystyle \text{Diameter}=\frac{\text{Circumference}}{\pi}$

Plug in the given circumference to find the diameter.

$\text{Diameter}=\frac{100\pi}{\pi}=100$ $\displaystyle \text{Diameter}=\frac{100\pi}{\pi}=100$

Now, recall the relationship between the diameter of the circle and the diagonal of the rectangle:

$\text{Diameter}=\text{Diagonal}=100$ $\displaystyle \text{Diameter}=\text{Diagonal}=100$

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Example Question #31 : How To Find The Length Of The Diagonal Of A Rectangle

If a rectangle is inscribed in a circle with a circumference of $16\pi$ $\displaystyle 16\pi$, what is the length of the diagonal of the rectangle?

Possible Answers:

$32\pi$ $\displaystyle 32\pi$

196 $\displaystyle 196$

$\displaystyle 4$

$\displaystyle 16$

Correct answer:

$\displaystyle 16$

Explanation:

You should notice that the diagonal of the rectangle is the same as the diameter of the circle.

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\text{diameter}\times\pi$ $\displaystyle \text{Circumference}=\text{diameter}\times\pi$

Since we are given the circumference and need the diameter, rerwite the equation to solve for the diameter.

$\text{Diameter}=\frac{\text{Circumference}}{\pi}$ $\displaystyle \text{Diameter}=\frac{\text{Circumference}}{\pi}$

Plug in the given circumference to find the diameter.

$\text{Diameter}=\frac{16\pi}{\pi}=16$ $\displaystyle \text{Diameter}=\frac{16\pi}{\pi}=16$

Now, recall the relationship between the diameter of the circle and the diagonal of the rectangle:

$\text{Diameter}=\text{Diagonal}=16$ $\displaystyle \text{Diameter}=\text{Diagonal}=16$

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Example Question #131 : Rectangles

Find the length of the diagonal of a rectangle with side lengths 6 and 7.

Possible Answers:

$\displaystyle 26$

$\sqrt{85}$ $\displaystyle \sqrt{85}$

$\displaystyle 42$

$\displaystyle 13$

Correct answer:

$\sqrt{85}$ $\displaystyle \sqrt{85}$

Explanation:

To find the diagonal of a rectangle recall that the diagonal will create a triangle where the width and length are legs of the triangle and the diagonal is the hypotenuse.

To solve, simple use the Pythagorean Theorem and solve for the hypotenuse, which will be the diagonal of the rectangle.

Thus,

$c=\sqrt{a^2+b^2}=\sqrt{36+49}=\sqrt{85}$ $\displaystyle c=\sqrt{a^2+b^2}=\sqrt{36+49}=\sqrt{85}$

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Example Question #542 : Plane Geometry

A rectangle has a width of $5\;cm$ $\displaystyle 5\;cm$ and a length that is $3\;cm$ $\displaystyle 3\;cm$ shorter than twice the width. What is the length of the diagonal rounded to the nearest tenth?

Possible Answers:

$7.9\;cm$ $\displaystyle 7.9\;cm$

$6.5\;cm$ $\displaystyle 6.5\;cm$

$8.6\;cm$ $\displaystyle 8.6\;cm$

$12.0\;cm$ $\displaystyle 12.0\;cm$

$11.2\;cm$ $\displaystyle 11.2\;cm$

Correct answer:

$8.6\;cm$ $\displaystyle 8.6\;cm$

Explanation:

First we must find the length of the rectangle before we can solve for the diagonal. With a length $3\;cm$ $\displaystyle 3\;cm$ shorter than twice the width, we can solve for length by drafting an algebraic equation:

$\\y=2x-3\;cm\\y=2(5\;cm)-3\;cm\\ y=10\;cm-3\;cm=7\;cm$ $\displaystyle \\y=2x-3\;cm\\y=2(5\;cm)-3\;cm\\ y=10\;cm-3\;cm=7\;cm$

Now that we know the values for length and width, we can use the Pythagorean Theorem to solve for the diagonal:

$\\c^2=5^2\;+7^2\;=25+49=74\\c=\sqrt{74}\approx8.6\;cm$ $\displaystyle \\c^2=5^2\;+7^2\;=25+49=74\\c=\sqrt{74}\approx8.6\;cm$

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Example Question #31 : How To Find The Length Of The Diagonal Of A Rectangle

Sarah's new tablet has a screen width of $7\;cm$ $\displaystyle 7\;cm$ and a length that is twice the width. What would be the area of the screen?

Possible Answers:

$78\;cm^2$ $\displaystyle 78\;cm^2$

$108\;cm^2$ $\displaystyle 108\;cm^2$

$84\;cm^2$ $\displaystyle 84\;cm^2$

$98\;cm^2$ $\displaystyle 98\;cm^2$

$89\;cm^2$ $\displaystyle 89\;cm^2$

Correct answer:

$98\;cm^2$ $\displaystyle 98\;cm^2$

Explanation:

If the width is $7\;cm$ $\displaystyle 7\;cm$ and the length is twice the width, we can calculate the area as such:

$\\area=length\cdot width \\area=2(7\;cm)\cdot7\;cm\\area=14\;cm\cdot7cm\\area=98\;cm^2$ $\displaystyle \\area=length\cdot width \\area=2(7\;cm)\cdot7\;cm\\area=14\;cm\cdot7cm\\area=98\;cm^2$

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Example Question #31 : How To Find The Length Of The Diagonal Of A Rectangle

Find the length of the diagonal of a rectangle with length 7 and width 2.

Possible Answers:

$\sqrt{53}$ $\displaystyle \sqrt{53}$

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

$3\sqrt2$ $\displaystyle 3\sqrt2$

$2\sqrt7$ $\displaystyle 2\sqrt7$

Correct answer:

$\sqrt{53}$ $\displaystyle \sqrt{53}$

Explanation:

To solve, simply use the Pythagorean Theorem. Thus,

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

$c=\sqrt{a^2+b^2}=\sqrt{7^2+2^2}=\sqrt{49+4}=\sqrt{53}$ $\displaystyle c=\sqrt{a^2+b^2}=\sqrt{7^2+2^2}=\sqrt{49+4}=\sqrt{53}$

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Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #22 : How To Find The Length Of The Diagonal Of A Rectangle

Example Question #23 : How To Find The Length Of The Diagonal Of A Rectangle

Example Question #24 : How To Find The Length Of The Diagonal Of A Rectangle

Example Question #24 : How To Find The Length Of The Diagonal Of A Rectangle

Example Question #25 : How To Find The Length Of The Diagonal Of A Rectangle

Example Question #31 : How To Find The Length Of The Diagonal Of A Rectangle

Example Question #131 : Rectangles

Example Question #542 : Plane Geometry

Example Question #31 : How To Find The Length Of The Diagonal Of A Rectangle

Example Question #31 : How To Find The Length Of The Diagonal Of A Rectangle

All Basic Geometry Resources

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