Geometry

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GRE Quantitative Reasoning › Geometry

Questions 1 - 10

What is the slope of a line which passes through coordinates $\dpi{100} \small (3,7)$ and $\dpi{100} \small (4,12)$ ?

$\dpi{100} \small 5$

$\dpi{100} \small \frac{1}{5}$

$\dpi{100} \small \frac{1}{2}$

$\dpi{100} \small 2$

$\dpi{100} \small 3$

Explanation

Slope is found by dividing the difference in the $\dpi{100} \small y$ -coordinates by the difference in the $\dpi{100} \small x$ -coordinates.

$\dpi{100} \small \frac{(12-7)}{(4-3)}=\frac{5}{1}=5$

Which of the above-listed lines are parallel?

, , and

None of them

and

All four lines

and

Explanation

There are several ways to solve this problem. You could solve all of the equations for . This would give you equations in the form . All of the lines with the same value would be parallel. Otherwise, you could figure out the ratio of to when both values are on the same side of the equation. This would suffice for determining the relationship between the two. We will take the first path, though, as this is most likely to be familiar to you.

Let's solve each for :

$y=2-\frac{2}{7}x$

$y=\frac{7}{2}-7x$

$y=\frac{4x}{-14}-\frac{91}{-14}$

$y=\frac{13}{2}-\frac{2x}{7}$

$y=\frac{814}{10.5}-\frac{3}{10.5}x$

Here, you need to be a bit more manipulative with your equation. Multiply the numerator and denominator of the value by :

$y=\frac{814}{10.5}-\frac{6}{21}x$

$y=\frac{814}{10.5}-\frac{2}{7}x$

Therefore, , , and all have slopes of $-\frac{2}{7}$

There is a line defined by the equation below:

There is a second line that passes through the point and is parallel to the line given above. What is the equation of this second line?

$y=-\frac{3}{4}x+2.75$

$y=-\frac{3}{4}x+1.25$

$y=\frac{3}{4}x+2.75$

$y=\frac{3}{4}x+1.25$

$y=\frac{3}{4}x+2.625$

Explanation

Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.

3x + 4y = 12

4y = _–_3x + 12

y = –(3/4)x + 3

slope = _–_3/4

We know that the second line will also have a slope of _–_3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.

y = mx + b

2 = _–_3/4(1) + b

2 = _–_3/4 + b

b = 2 + 3/4 = 2.75

Plug the y-intercept back into the equation to get our final answer.

y = –(3/4)x + 2.75

For $\$15$ , Chelsea can get either a $16\ in$ diameter pizza or two $8\ in$ diameter pizzas. Which is the better deal?

$16\ in$

two $8\ in$

The two values are equal.

Cannot be determined.

Explanation

$A=\pi r^2$

$A_{16\ in}=8^2\pi =64\pi, \ A_{8\ in}=4^2\pi =16\pi$

$2(A_{8\ in})=2(16\pi )=32\pi$

Therefore the 16 inch pizza is the better deal.

Circlechord0

O is the center of the circle above.

The length of is .

Quantity A: The area of the circle.

Quantity B: $49\pi$

Which of the following is true?

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined.

Explanation

Circlechord0

O is the center of the circle above.

The length of is .

Quantity A: The area of the circle.

Quantity B: $49\pi$

Do not be tricked by this question. It is true that can be split into halves, each of which are in length. These halves are not, however, radii to the circle. Since this does not go through the center of the circle, its length is shorter than the diameter. This means that the radius of the circle must be greater than . Now, if it were , the area would be $49\pi$ . Since it is larger than , the area must be larger than $49\pi$ . Quantity A is larger than quantity B.

Circlechord1

O is the center of the circle above.

The circumference of the circle above is $30\pi$ .

Quantity A: The length of .

Quantity B:

Which of the following is true?

Quantity B is larger.

Quantity A is larger.

The relationship cannot be determined.

The two quantities are equal.

Explanation

Now, we know that the circumference of a circle is:

$C=2\pi r$ or $C=\pi d$

This means that the diameter of our circle is must be . Given this, we know that the must be shorter than , for the diameter is the longer than any chord that does not pass through the center of the circle. Quantity B is larger than quantity A.

Which of the following lines is perpendicular to the line ?

$\frac{y}{3}-x=3$

Explanation

Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.

The slope of this line is . The negative reciprocal will be $+\frac{1}{3}$ , which will be the slope of the perpendicular line.

Now we need to find the answer choice with this slope by converting to slope-intercept form.

$y=\frac{4}{12}x-\frac{6}{12}$

$y=\frac{1}{3}x-\frac{1}{2}$

This equation has a slope of $+\frac{1}{3}$ , and must be our answer.

The equation of a particular line is as follows:

$y=\frac{-7}{3}x +12$ .

What is the slope of a line perpendicuar to this one?

$\frac{3}{7}$

$\frac{7}{3}$

$\frac{-3}{7}$

Explanation

The slope of a perpendicular line is the opposite reciporical of the originial slope.

The opposite would be changing the sign from positive to negative or vice versa.

The reciporical is flipping the fraction or switching the numerator and denominator.

This equation is in slope-intercept form,

meaning m represents the slope of the equation.

Since the original is $\frac{-7}{3}$ , then the perpendicular would be positive and would swtich the 7 and 3 giving us $\frac{3}{7}$ .

An obtuse Isosceles triangle has two sides with length and one side length . The length of side $\frac{3}{4}$ ft. If the length of half the length of side , what is the perimeter of the triangle?

$1\frac{1}{2}$ ft

$\frac{12}{4}$ ft

$1\frac{2}{3}$ ft

$1\frac{1}{3}$ ft

Explanation

By definition, an Isosceles triangle must have two equivalent side lengths. Since we are told that $b=\frac{3}{4}$ ft and that the sides with length are half the length of side , find the length of by: $\frac{3}{4}=\frac{6}{8}$ and half of $\frac{6}{8}=\frac{3}{8}$ . Thus, both of sides with length must equal $\frac{3}{8}$ ft.