GRE Math : Plane Geometry

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Example Questions

Example Question #1 : How To Find The Length Of The Diagonal Of A Square

If a square has a side length of 4, how long is the diagonal of the square?

Possible Answers:

4√2

2√2

8√2

Correct answer:

4√2

Explanation:

A diagonal divides a square into two 45-45-90 triangles, which have lengths adhering to the ratio of x: x: x√2. Therefore, 4√2 is the correct answer as the diagonal represents the hypotenuse of the triangle. the Pythagorean theorem can also be used: 4²+ 4² = c^2.

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Example Question #2 : How To Find The Length Of The Diagonal Of A Square

If a square has a side length of √2, how long is the diagonal of the square?

Possible Answers:

4-√2

2√2

4√2

Correct answer:

Explanation:

A diagonal divides a square into two 45-45-90 triangles, which have lengths adhering to the ratio of x: x: x√2. Therefore, 2 is the correct answer as the diagonal represents the hypotenuse of the triangle. the Pythagorean theorem can also be used: √2²+ √2² = c^2.

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Example Question #3 : How To Find The Length Of The Diagonal Of A Square

A square has a side of length 5. What is the length of its diagonal?

Possible Answers:

5√3

10√3

5√2

Correct answer:

5√2

Explanation:

The diagonal separates the square into two 45-45-90 right triangles. The problem can be solved by using the Pythagorean Theorem, a²+ b² = c².It can also be solved by recognizing the 45-45-90 special triangles, which have side ratios of x : x : x√2.

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Example Question #61 : Quadrilaterals

A square with width of $6\ m$ is inscribed in a circle. What is the total area inside the circle?

Possible Answers:

$64\pi$

$81\pi$

$18\pi$

$49\pi$

$100\pi$

Correct answer:

$18\pi$

Explanation:

We know that each side of the square is 6, so use the Pythagorean Theorem to solve for the diagonal of the square. The diagonal of the square is also the diameter of the circle.

$6^{2}+6^{2} = d^{2}$

$72= d^{2}$

$\dpi{100} \small \sqrt{72}=d$

$\dpi{100} \small \sqrt{72}=\sqrt{36\times 2}=6\sqrt{2}$

$\dpi{100} \small 6\sqrt{2}=d$

Therefore the radius must be $\dpi{100} \small r=\frac{6\sqrt{2}}{2}\rightarrow 3\sqrt{2}$

Now let's find the area inside the circle using the radius.

$\dpi{100} \small A=\pi r^{2}$

$\dpi{100} \small A=\pi\left (3\sqrt{2} \right )^{2}$

$\dpi{100} \small A=\pi\left (3\sqrt{2} \right )^{2}\rightarrow \pi\left ( 9\times 2 \right )\rightarrow 18\pi$

$\dpi{100} \small A=18\pi$ meters²

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Example Question #1 : How To Find The Length Of The Diagonal Of A Square

Quantity A:

The diagonal of a square with a side-length of .

Quantity B:

The side-length of a square with a diagonal of .

Possible Answers:

The relationship between the two quantities cannot be determined from the information given.

Both quantities are equal.

Quantity A is greater.

Quantity B is greater.

Correct answer:

Both quantities are equal.

Explanation:

Quantity A: The diagonal of a square with a side-length of 7.

Quantity B: The side-length of a square with a diagonal of 14.

Both quantities can be determined, so can the relationship.

Quantity A:

To determine the diagonal of a square, it's important to remember that a diagonal directly bisects a square from corner to corner. In other words, it bisects the corners, creating two triangles with 45:45:90 proportions, with the diagonal serving as the hypotenuses . If you remember your special triangles, then the side-side-hypotenuse measurements have a ratio of $1:1:\sqrt{2}$ .

In Quantity A, the side-length is 7. Following the proportion:

$1(7):1(7): \sqrt{2}(7)$

The diagonal equals $7\sqrt{2}$ .

Quantity B:

We can use the same ratio to figure out quantity B by substituting x for the unknown side-length quantity, which looks like this:

$1(x):1(x):\sqrt{2}(x) = 14$

To find x in this ratio, just isolate x in the hypotenuse:

$\sqrt{2}x = 14$

Divide by $\sqrt{2}$

$x = \frac{14}{\sqrt{2}}$

Now, how does Quantity A and Quantity B match up?

$\frac{14}{\sqrt{2}} \approx9.89$

$7\sqrt{2}\approx9.89$

On the surface, it looks like the two quanties are equal. But how do we prove it? Well, we know that $\sqrt{2}\times \sqrt{2} = 2$ . Therefore, we know that:

$7\sqrt{2}\times\sqrt{2}=14$

Divide both sides by $\sqrt{2}$

$7\sqrt{2} = \frac{14}{\sqrt{2}}$

Therefore, both quantities are equal.

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Example Question #1 : How To Find The Length Of The Side Of A Square

Quantity A:

The side-length of a square with a perimeter of .

Quantity B:

The side-length of a square with an area of .

Possible Answers:

Both quantities are equal

Quantity B is greater

Quantity A is greater

The relationship cannot be determined from the information given

Correct answer:

Quantity A is greater

Explanation:

The first step to a quantitative comparison is to determine whether it can be solved at all with the given knowledge. Since all you need to find the side-length of a square is the perimeter, the area, OR the diagonal and we have one of each for these two quantities, this relationship can be determined. Thus, "the relationship cannot be determined" is out.

Now, to solve both quantities. Quantity A can be solved by translating the perimeter into side lengths: the formula for the perimeter of a square is P = 4s , with being the side-length, so you just need to divide the perimeter by four.

30=4s

$\frac{30}{4}=s$

Thus, quantity A is 7.5 .

Quantity B can be solved by translating the area into side lengths: the formula for the area of a square is $A=l \times w$ , or $A=s^{2}$ , with being the side-length, so you just need to find the square root of the area.

30=s^2

$\sqrt30=s$

Thus, quantity B is roughly 5.48 .

Therefore quantity A is greater.

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Example Question #184 : Geometry

Inscribedsquare

Circle has a center in the center of Square ABCD .

If the area of Circle is $324\pi$ , what is the length of ?

Possible Answers:

162

324

Correct answer:

Explanation:

If the area of Circle is $324\pi$ , we know that the area can be computed using the standard area formula:

$A = \pi r^2$ , using $324\pi$ for

$324\pi = \pi r^2$

Simplifying, we get:

324=r^2

We know that must be less than . By choosing for , we find out that this is the radius of our circle. Thus, we know that the diameter of the circle is double this, or . Now, consider the following diagram:

Inscr18

Notice that the diameter is the same length as a side of the square. Thus, is equal to .

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Example Question #1 : How To Find The Length Of The Side Of A Square

Prb11 1

Circle has a center in the center of square ABCD .

The line segment marked with length lies on the diagonal of the square ABCD .

What is the length of side ?

Possible Answers:

This cannot be computed from the given data

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

$\frac{10}{\sqrt{2} - 1}$

$\frac{3\sqrt{2}-4}{5}$

$\sqrt{2}+7$

Correct answer:

$\frac{10}{\sqrt{2} - 1}$

Explanation:

You can further fill in your diagram as follows:

Prb11 2

Now, we know that the triangle BCD is a 45-45-90 triangle. We also know that the length of and must also be equal to the diameter of the circle. (The diameter of the circle will run across the circle horizontally if you draw it that way. This will provide you with a complete side length.) Now, we know that the ratio of the hypotenuse of BCD to the side must be the same as:

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

For our data, that means:

$\frac{5+5+d}{d} = \frac{\sqrt{2}}{1}$

Simplifying, we know:

$\frac{10+d}{d} = \frac{\sqrt{2}}{1}$

$\frac{10}{d}+1 = \sqrt{2}$

$\frac{10}{d} = \sqrt{2} - 1$

Now, make both sides reciprocals:

$\frac{d}{10} = \frac{1}{\sqrt{2} - 1}$

Finally, solve:

$d = \frac{10}{\sqrt{2} - 1}$

Recall, this is both the length of the side and the diameter of the circle. Hence, you have your answer.

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GRE Math : Plane Geometry

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #1 : How To Find The Length Of The Diagonal Of A Square

Example Question #2 : How To Find The Length Of The Diagonal Of A Square

Example Question #3 : How To Find The Length Of The Diagonal Of A Square

Example Question #61 : Quadrilaterals

Example Question #1 : How To Find The Length Of The Diagonal Of A Square

Example Question #1 : How To Find The Length Of The Side Of A Square

Example Question #184 : Geometry

Example Question #1 : How To Find The Length Of The Side Of A Square

All GRE Math Resources

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