Coordinate Geometry - SAT Subject Test in Math I

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Question

Find the distance between and .

Answer

For this problem we will need to use the distance formula:

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

In our case,

and .

Plugging these values into the formula we are able to find the distance.

$d = \sqrt {(3-(-12))^2 + (4-(-5)))^2))}$

$=\sqrt {15^2+9^2}$

$= \sqrt {306}$

$= \sqrt {9 \cdot 34} = 3 \sqrt {34}$

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Question

What is the distance between and ?

Answer

Write the distance formula.

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Substitute the points:

$d=\sqrt{(-3-1)^2+(-2-6)^2}$

$d=\sqrt{16+64} = \sqrt{80}$

The radical can be broken down into factors of perfect squares.

$\sqrt{80} = \sqrt{16}\cdot \sqrt5 = 4\sqrt5$

The answer is: $4\sqrt5$

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Question

What is the midpoint of the points (3,12) and (9,15)?

Answer

To find the midpoint we must know the midpoint formula which is

$(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$

We then take the -coordinate from the first point and plug it into the formula as $x_{1}$ .

We take the -coordinate from the second point and plug it into the formula as $x_{2}$ .

We then do the same for $y_{1}$ and $y_{2}$ .

With all of the points plugged in our equation will look like this.

$(\frac{3+9}{2},\frac{12+15}{2})$

We then perform the necessary addition and division to get the answer of

$(\frac{12}{2},\frac{27}{2})=(6,13.5)$

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Question

Find the midpoint of the line segment that connects the two points below.

Point 1:

Point 2:

Answer

The average of the the -coordinates and the average of the y-coordinates of the given points will give you the mid-point of the line that connects the points.

$\textup{midpoint}=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$ , where is and is .

$(\frac{(0)+(-2)}{2}, \frac{(-3)+(4)}{2})$

$(\frac{-2}{2},\frac{1}{2})$

$(-1, \frac{1}{2})$

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Question

Find the midpoint of the line segment with endpoints and .

Answer

Use the midpoint formula:

$\left ( \frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}} \right )$

Substitute:

$\frac{x_{1}+x_{2}}{2} =\frac{3.6+7.2}{2} = \frac{10.8}{2} = 5.4$

$\frac{y_{1}+y_{2}}{2}} \right = \frac{9.1+(-1.5)}{2}} = \frac{7.6}{2} = 3.8$

The midpoint is

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Question

What is the midpoint between $\left ( 2,3\right )$ and $\left ( -7,4\right )$ ?

Answer

The formula to find the midpoint is as follows:

$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

In our case our

our

substituting in these values we get

midpoint = $\left(\frac{2+-7}{2}, \frac{3+4}{2}\right)=\left(\frac{-5}{2}, \frac{7}{2}\right)$

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Question

Find the midpoint of the line that passes through the points and .

Answer

Recall the midpoint formula as $(\frac{x1+x2}{2}, \frac{y1+y2}{2})$ .

Thus,

$(\frac{-1+5}{2},\frac{4+2}{2}) = (2,3)$

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Question

Find the midpoint of the line that passes through the points and .

Answer

Recall the midpoint formula as $(\frac{x1+x2}{2}, \frac{y1+y2}{2})$ .

Thus,

$(\frac{2+(-2)}{2},\frac{-1+1}{2}) = (0,0)$

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Question

Find the midpoint of the line segment with these endpoints: and .

Answer

The midpoint formula is

$\left(\frac{x_{1}+x_{2}}2,\frac{y_{1}+y_2}{2}\right)$ .

Add the two values together and divide the total by two to find the value of the midpoint. Add the two values together and divide the total by two to find the value of the midpoint.

$\ \left(\frac{5+9}2,\frac{7+15}{2}\right)\ \=\left(\frac{14}{2},\frac{22}{2} \right )\ \=(7,11)$

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Question

What is the midpoint of a line that connects the points and ?

Answer

The midpoint formula is $\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)$ .

Therefore, all we need to do is plug in the points given to us to find the midpoint.

In this case, $x_{1}=1,x_{2}=-7,y_{1}=5,y_{2}=9$ .

To find the -value for the midpoint, we add: . Then we divide by : .

To find the -value for the midpoint, we add: . Then we divide by : .

So our solution is .

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Question

Find the midpoint of a line with the endpoings (3, 4) and (-1, -1).

Answer

When finding the midpoint between two points, we use the midpoint formula

$\left(\frac{x_1 + x_2}{2}, \frac{y_1+y_2}{2}\right)$

where and are the points given.

Knowing this, we can substitute the values into the formula. We get

$\left(\frac{3 + -1}{2}, \frac{4 + - 1}{2}\right)$

$\left(\frac{3-1}{2}, \frac{4-1}{2}\right)$

$\left(\frac{2}{2}, \frac{3}{2}\right)$

$\left(1, \frac{3}{2}\right)$

Therefore, $\left(1, \frac{3}{2}\right)$ is the midpoint.

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Question

A line is connected by points and on a graph. What is the midpoint?

Answer

Write the midpoint formula.

$(x,y) = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

Let and .

Substitute the given points.

$(x,y) = (\frac{2+(-1)}{2}, \frac{5+3}{2})$

Simplify the coordinate.

$(x,y) =(\frac{1}{2},4)$

The answer is: $(\frac{1}{2},4)$

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Question

What is the midpoint between $\left ( 1,3\right )$ and $\left ( 5,9\right )$ ?

Answer

To find the midpoint of two points you find the average of both the values and values.

For ,

$\frac{1+5}{2}=3$ .

For ,

$\frac{3+9}{2}=6$ .

This means the midpoint is $\left ( 3,6\right )$ .

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Question

Find the midpoint between (2,8) and (8,6).

Answer

The midpoint formula says that:

$Midpoint=(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$

Given (2,8) and (8,6); $\small x_{1}=2$ $\small \small x_{2}=8$ $\small \small y_{1}=8$ $\small \small y_{2}=6$

Plug the values into the formula and reduce:

$(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})=(\frac{2+8}{2},\frac{8+6}{2})=(5,7)$

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Question

Find the midpoint between and .

Answer

Write the formula for the midpoint.

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

Substitute the point into the formula.

$M = (\frac{-1+3}{2},\frac{-3+(-1)}{2} ) = (1,-2)$

The answer is:

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Question

Give the axis of symmetry of the parabola of the equation

$f(x) = 3x^{2} - 9x + 16$

Answer

The line of symmetry of the parabola of the equation

$f(x) = ax^{2}+ bx + c$

is the vertical line

$x = -\frac{b}{2a}$

Substitute :

The line of symmetry is

$x = -\frac{-9}{2 \cdot 3} = \frac{9}{6} = \frac{3}{2}$

That is, the line of the equation $x = \frac{3}{2}$ .

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Question

What is the center of the circle with the following equation?

Answer

Remember that the basic form of the equation of a circle is:

This means that the center point is defined by the two values subtracted in the squared terms. We could rewrite our equation as:

Therefore, the center is

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Question

What is the area of the sector of the circle formed between the -axis and the point on the circle found at when the equation of the circle is as follows?

Round your answer to the nearest hundreth.

Answer

For this question, we will need to do three things:

Determine the point in question.

Use trigonometry to find the area of the angle in question.

Use the equation for finding a sector area to finalize our answer.

Let us first solve for the coordinate by substituting into our equation:

Our point is, therefore:

Now, we need to calculate the angle formed between the origin and the point that we were given. We can do this using the inverse tangent function. The ratio of to is here: $\frac{8}{6}$

Therefore, the angle is:

$\tan^{-1}(\frac{8}{6})=53.13010235415592$

To solve for the sector area, we merely need to use our standard geometry equation. Note that the radius of the circle, based on the equation, is .

$\frac{53.13010235415592}{360}\pi10^2 = 46.36476090008$

This rounds to .

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Question

What is the area of the sector of the circle formed between the -axis and the point on the circle found at when the equation of the circle is as follows?

Round your answer to the nearest hundreth.

Answer

For this question, we will need to do three things:

Determine the point in question.

Use trigonometry to find the area of the angle in question.

Use the equation for finding a sector area to finalize our answer.

Let us first solve for the coordinate by substituting into our equation:

Our point is, therefore:

Now, we need to calculate the angle formed between the origin and the point that we were given. We can do this using the inverse tangent function. The ratio of to is here: $\frac{10.82774214691133}{12.4}$

Therefore, the angle is:

$\tan^{-1}(\frac{10.82774214691133}{12.4})=41.1276246899636$

To solve for the sector area, we merely need to use our standard geometry equation. Note that , based on the equation, is .

$\frac{41.1276246899636}{360}\pi271 = 97.2635889213762$

This rounds to .

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Question

If the center of a circle is at $\left ( 0,0\right )$ and it has a radius of , what positive point on the does it intersect?

Answer

Since you are looking for a point on the , your value will be zero.

The center of the circle is at the origin and radius is the distance from the center, so that means the point you are looking for must be points away from .

This can be two points on the but since you are looking for a positive one, your answer must be $\left ( 0,4\right )$ .

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