Coordinate Geometry - GMAT Quantitative Reasoning

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Question

The midpoint of a line segment with endpoints and is . What is ?

Answer

If the midpoint of a line segment with endpoints and is , then by the midpoint formula,

$\frac{2A + (B + 7)}{2} = 9$

and

$\frac{B + (3A - 2)}{2} = 6$ .

The first equation can be simplified as follows:

or

The second can be simplified as follows:

or

This is a system of linear equations. can be calculated by subtracting:

$\underline{2A + B = 11 }$

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Question

The quadrilateral with vertices is a trapezoid. What are the endpoints of its midsegment?

Answer

The midsegment of a trapezoid is the segment whose endpoints are the midpoints of its legs - its nonparallel opposite sides. These two sides are the ones with endpoints and . The midpoint of each can be found by taking the means of the - and -coordinates:

$(0,0), (4,4): \left ( \frac{0+4}{2},\frac{0+4}{2} \right ) \Rightarrow (2,2)$

$(22,0), (16,4): \left ( \frac{22+16}{2},\frac{0+4}{2} \right ) \Rightarrow (19,2)$

The midsegment is the segment that has endpoints (2,2) and (19,2)

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Question

If the midpoint of $\overline{VU}$ is $\left ( 6,7\right )$ and $\textup{Point }V$ is at $\left (2,9\right )$ , what are the coordinates of $\textup{Point }U$ ?

Answer

Midpoint formula is as follows:

$\small \small m(x,y)=(\frac{x+x'}{2},\frac{y+y'}{2})$

In this case, we have x,y and the value of the midpoint. We need to findx' and y'

V is at (2,9) and the midpoint is at (6,7)

$\small 6=\frac{2+x'}{2}$ and $\small 7=\frac{9+y'}{2}$

$\small x'=62-2=10$ $\small y'=72-9=14-9=5$

So we have (10,5) as point U

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Question

A line segment has its midpoint at and an endpoint at . What are the coordinates of the other endpoint?

Answer

Because we are given the midpoint and one of the endpoints, we know the x coordinate of the other endpoint will be the same distance away from the midpoint in the x direction, and the y coordinate of the other endpoint will be the same distance away from the midpoint in the y direction. Given two endpoints of the form:

The midpoint of these two endpoints has the coordinates:

$(x_m,y_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Plugging in values for the given midpoint and one of the endpoints, which we can see is because it lies to the right of the midpoint, we can solve for the other endpoint as follows:

$(x_m,y_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$(-1,3)=(\frac{x_1+3}{2},\frac{y_1+8}{2})$

$\frac{x_1+3}{2}=-1\rightarrow x_1+3=-2\rightarrow x_1=-5$

$\frac{y_1+8}{2}=3\rightarrow y_1+8=6\rightarrow y_1=-2$

So the other endpoint has the coordinates

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Question

Consider segment with endpoint at . If the midpoint of can be found at , what are the coordinates of point ?

Answer

Recall midpoint formula:

$(x',y')=\left(\frac{x^1+x^2}{2}, \frac{y^1+y^2}{2}\right)$

In this case we have (x'y') and one of our other (x,y) points.

Plug and chug:

$(-15,-67)=\left(\frac{8+x}{2}, \frac{9+y}{2}\right)$

If you make this into two equations and solve you get the following.

$\small -15=\frac{8+x}{2}, x=-38$

$\small -67=\frac{9+y}{2}, y=-143$

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Question

In the -plane, point $(a,b)$ lies on a circle with center at the origin. The radius of the circle is 5. What is the value of $a^{2}+b^{2}$ ?

Answer

$a$ and $b$ are the right-angle sides of a triangle, and the radius of the circle is the hypotenuse of the triangle. From the Pythagorean Theorem we would know that $a^{2}+b^{2}=r^{2}=5^{2}=25$ .

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Question

Describe the circle given by the equation $(x+3)^{2}+y^{2} = 2$ .

Answer

The equation for a circle is $(x-a)^{2}+(y-b)^{2} = r^{2}$ , where (a, b) is the center and r is the radius. In our equation, a = –3, b = 0, and r = $\sqrt{2}$ . Then the equation describes a circle with a center at (–3, 0) and a radius of $\sqrt{2}$ .

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Question

One of the diameters of a circle has endpoints (4, 5) and (10, 1). What is the equation of this circle?

Answer

The equation of a circle with center and radius is

$\small \small \small \small (x - h)^{2} + (y - k)^{2} = r^{2}$

The center is the midpoint of any diameter, so to find the center, we use the midpoint formula:

$\small \small h= \small \frac{x_{1}+x_{2}}{2}= \small \frac{4+10}{2} = 7$

$\small \small \small k= \small \frac{y_{1}+y_{2}}{2}= \small \frac{5+1}{2} = 3$

The center is (7,3). The radius is the distance between (7,3) and (10,1), so we use the distance formula:

$\small r = \sqrt{(10-7)^{2}+(1-3))^{2}} = \sqrt{3^{2}+(-2)^{2}} = \sqrt{13}$

So $\small r^{2} = 13$ , and the equation of the circle is

$\small (x - 7)^{2} + (y - 3)^{2} = 13$

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Question

In the coordinate plane, a circle has center $\left ( 5,2 \right )$ and passes through the point $\left ( -1,-6 \right )$ . What is the area of the circle?

Answer

The distance of the two points is $\sqrt{(5-(-1))^2+(2-(-6))^2}=10$ .

So 10 is the radius of the circle. Then we can calculate the area:

$A=\pi r^{2}=\pi \cdot 10^{2}=100\pi$ .

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Question

A circle on the coordinate plane has area $64\pi$ ; its center is the origin. Which of the following is the equation of this circle?

Answer

The equation of a circle with center at the origin is

$x ^{2} + y ^{2} = r^{2}$

where is the radius of the circle. The area of the circle is $A = \pi r ^{2}$ .

Since the area of the circle in the question is $64\pi$ , we can solve for $r ^{2}$ :

$A = \pi r ^{2} = 64 \pi$

$\pi r ^{2} \div \pi = 64 \pi \div \pi$

$r ^{2} = 64$

The equation is

$x ^{2} + y ^{2} = 64$

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Question

$\textup{Circle }Q$ is centered at the point $\left ( -6,-5\right )$ and touches the y-axis once at the point $\left ( 0,-5\right )$ . What is the equation for $\textup{Circle }Q$ ?

Answer

The equation of a circle in gneral form is:

$\small r^2=(x-h)^2+(y-k)^2$

In this case, our radius is 6, because our circle touches the y-axis once at the point (0,-5). This makes our radius equal to the absolute value of the x-coordinate of the center of our circle. Eliminate anything that doesn't have a 36.

Then, because our (h,k) are already negative, they change the signs to positive within the parentheses making our answer:

$\small 36=(x+6)^2+(y+5)^2$

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Question

The points and form a line which passes through the center of circle Q. Both points are on circle Q.

Which of the following represent the correct equation for circle Q?

Answer

The general form for an equation of a circle is as follows:

Where r is the radius, and (h,k) are the coordinates of the center of the circle.

To begin, let's find the radius using distance formula. Because LK passes through the center of the circle and goes from the outer edge of the circle to the other side, we can say that LK is our diameter.

Use distance formula to find the length of LK.

$d=\sqrt{(x-x')^2+(y-y')^2}$

Plug in our points and simplify:

$d=\sqrt{(9--4)^2+(3-5)^2}=\sqrt{13^2+2^2}=\sqrt{173}\approx13.15$

That is our diameter, so our radius will be half of 13.15, or 6.575. This rounds to 6.58

Next, we can use midpoint formula to find the center of circle Q. Midpoint formula is:

$midpoint(x,y)=\left(\frac{x+x'}{2},\frac{y+y'}{2}\right)$

Plug in and simplify to find our midpoint

$midpoint(x,y)=\left(\frac{-4+9}{2},\frac{5+3}{2}\right)=\left(\frac{5}{2},\frac{8}{2}\right)=(2.5,4)$

Put it all together to get:

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Question

Find the equation of a circle whose radius is and whose center is $\left (1,2 \right )$ .

Answer

To solve this problem, remember that the general formula for a circle with center $\left ( h,k \right )$ and radius is:

Therefore,

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Question

Find the equation of circle whose radius is and center is at .

Answer

To solve, simply use the formula given center at $\small (h, k)$ and radius of $\small r$ .

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Question

Two circles on the coordinate plane have the origin as their center. The outer circle has as its equation

$x^{2}+ y^2 = 64$ ;

the region between the circles has area $30 \pi$ .

Give the equation of the inner circle.

Answer

A circle with its center at the origin has as its equation

$x^{2}+ y^2 = r^{2}$ .

Since the equation of the outer circle is $x^{2}+ y^2 = 64$ , then, for this circle,

$r^{2} = 64$ ,

and its area is $A = \pi r^{2} = \pi \cdot 64 = 64 \pi$ .

The area of the region between the circles is $30 \pi$ , so the inner circle has area

$64 \pi - 30 \pi = 34 \pi$ .

If is the radius of the inner circle, then its area is

$A = \pi R^{2} = 34\pi$

This makes $R ^{2} = 34$ , and the equation of the inner circle

$x^{2}+ y^2 = R^{2}$

or

$x^{2}+ y^2 =34$

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Question

Two circles on the coordinate plane have the origin as their center. The inner circle has as its equation

$x^{2}+ y^2 = 64$ ;

the region between the circles has area $30 \pi$ .

Give the equation of the outer circle.

Answer

A circle with its center at the origin has as its equation

$x^{2}+ y^2 = r^{2}$ .

Since the equation of the inner circle is $x^{2}+ y^2 = 64$ , then, for this circle,

$r^{2} = 64$ ,

and its area is $A = \pi r^{2} = \pi \cdot 64 = 64 \pi$ .

The area of the region between the circles is $30 \pi$ , so the outer circle has area

$64 \pi + 30 \pi = 94 \pi$ .

If is the radius of the outer circle, then its area is

$A = \pi R^{2} = 94\pi$

This makes $R ^{2} = 94$ , and the equation of the outer circle

$x^{2}+ y^2 = R^{2}$

or

$x^{2}+ y^2 =94$

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Question

Two circles on the coordinate plane have the origin as their center. The outer circle has as its equation

$x^{2}+ y^2 = 50$ ;

the inner circle has as its equation

$x^{2}+ y^2 = 25$ .

Give the area of the region between the two circles.

Answer

A circle with its center at the origin has as its equation

$x^{2}+ y^2 = r^{2}$ .

Let and be the radii of the larger and smaller circles, respectively.

The larger circle has equation $x^{2}+ y^2 = 50$ , so $R^{2} = 50$ . The area of a circle is equal to $\pi$ times the square of the radius, so the area of the larger circle is $\pi \cdot R^{2} = 50 \pi$ .

Similarly, the area of the smaller circle, whose equation is $x^{2}+ y^2 = 25$ , is $25 \pi$ .

The area of the region between them is the difference, which is $50 \pi - 25 \pi = 25 \pi$ .

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Question

Two circles on the coordinate plane have the origin as their center. The outer circle has twice the circumference as the inner circle, the equation of which is

$x^{2}+ y^{2} = 16$ .

Give the equation of the outer circle.

Answer

The equation of a circle centered at the origin is

$x^{2}+ y^{2} = r^{2}$

where is the radius of the circle. Since the equation of the outer circle is

$x^{2}+ y^{2} = 16$ ,

$r^{2} = 16$ ,

and the radius is the square root of this:

.

The circumference of this circle is $2 \pi$ times this, or

$C= 2 \pi \cdot 4 = 8 \pi$ .

The circumference of the larger circle is twice this, or $2 \cdot 8 \pi = 16 \pi$ ; divide this by $2 \pi$ to get the radius of the larger circle, which is

$R = 16 \pi \div 2 \pi = 8$ .

Consequently, $R ^{2} = 8 ^{2} = 64$ .

The equation of the larger circle is

$x^{2}+ y^{2} = R^{2}$ , or

$x^{2}+ y^{2} = 64$ .

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Question

Two circles on the coordinate plane have the origin as their center. The outer circle has twice the area as the inner circle, the equation of which is

$x^{2}+ y^{2} = 5$ .

Give the equation of the outer circle.

Answer

The equation of a circle centered at the origin is

$x^{2}+ y^{2} = r^{2}$

where is the radius of the circle. Since the equation of the outer circle is

$x^{2}+ y^{2} = 5$ ,

$r^{2} = 5$ ,

and the area is

$A = \pi r^{2} = \pi \cdot 5 = 5 \pi$ .

The area of the larger circle is twice this, or $2 \cdot 5 \pi = 10 \pi$ ; that is,

$A_{2} = \pi R^{2} = 10 \pi$

$R^{2} = 10$ ,

and the equation of that outer circle is

$x^{2}+ y^{2} = 10$ .

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Question

Two circles on the coordinate plane have the origin as their center. The outer circle has area five times that of the inner circle; the region between them has area $20 \pi$ . Give the equation of the inner circle.

Answer

Let be the radius of the inner circle. The area of the inner circle is $A = \pi r^{2}$ ; the outer circle has area five times this, or $A '= 5 \pi r^{2}$ ; the region between them has area equal to the difference of these quantities, or

$A'' = A ' - A= 5 \pi r^{2} - \pi r^{2} = 4 \pi r^{2}$

This is equal to $20 \pi$ , so

$4 \pi r^{2} = 20 \pi$

$\frac{4 \pi r^{2} }{4 \pi}= \frac{20 \pi}{4 \pi }$

$r^{2} =5$

A circle with its center at the origin has as its equation

$x^{2}+ y^2 = r^{2}$ ,

so the inner circle has as its equation

$x^{2}+ y^2 = 5$ .

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