Coordinate Geometry - GED Math

Card 0 of 710

What is the x-intercept of the following equation?

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The x-intercept is the value of when . Substitute the value for y.

Add one on both sides.

Divide by three on both sides.

$\frac{1}{3}$=\frac{3x}{3}$

The answer is: $\frac{1}{3}$

Use the distance formula to calculate the distance between the points $\small \small (4,2)$ and $\small (-7,3)$ .

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The distance between 2 points can be determined using the distance formula:

$\small D=$\sqrt{(x_1-x_$2)^2$+(y_1-y_$2)^2$}$$

$\small $D=$\sqrt{(4-(-7))^2$$+(2-3)^2$}$$

$\small $D=$\sqrt{(11)^2$$+(-1)^2$}$$

$\small D=$\sqrt{121+1}$=$\sqrt{122}$$

Provide your answer in its most simplified form.

Find the distance between the following two points:

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To find the distance we need to use the distance formula:

$d = $$\sqrt{(x_{1}$$ - x_${2})^{2}$ + $(y_{1}$ - y_${2})^{2}$}$

Plug in your x and y values to get:

$d = $\sqrt{(7 - $(-3))^{2}$$ + (0 - $5)^{2}$}$

Combine like terms to get:

$d = $$\sqrt{(10)^{2}$$ + $(-5)^{2}$}$

Continue with your order of operations:

$d = $\sqrt{100 + 25}$$

$d = $\sqrt{125}$$

Don't forget to simplify if possible:

$d = 5$\sqrt{5}$$

Provide your answer in its most simplified form.

Find the distance between these two points:

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For this problem we must use the distance formula:

$d = $$\sqrt{(x_{1}$$ - x_${2})^{2}$ + $(y_{1}$ - y_${2})^{2}$}$

Plug in your x and y values:

$d = $\sqrt{((-10) - $3)^{2}$$ + ((-4) - $8)^{2}$}$

Combine like terms:

$d = $$\sqrt{(-13)^{2}$$ + $(-12)^{2}$}$

Continue your order of operations:

$d = $\sqrt{169 +144}$$

$d = $\sqrt{313}$$

This cannot be simplified so you are left with the correct answer.

Provide your answer in its most simplified form.

Find the distance between the two following points:

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We must use the distance formula to solve this problem:

$d = $$\sqrt{(x_{1}$$ - x_${2})^{2}$ + $(y_{1}$ - y_${2})^{2}$}$

Plug in your x and y values:

$d = $\sqrt{(2 - $5)^{2}$$ + (2 - $6)^{2}$}$

Combine like terms:

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

Continue with your order of operations

$d = $\sqrt{9 + 16}$$

$d = $\sqrt{25}$$

Simplify to get:

What is the distance between the points and ?

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Write the distance formula.

$D=$\sqrt{(x_2-x_$1)^2$+(y_2-y_$1)^2$}$$

Substitute the points into the formula.

Factor the radical using factors of perfect squares.

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

The answer is: $4$\sqrt{5}$$

What is the distance between the points and ?

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Remember that you can consider your two points as:

and

From this, remember that the distance formula is:

$\sqrt{(x_1-x_$2)^2$+(y_1-y_$2)^2$}$

Now, for your data, this is:

or

You can simplify this value a little. Identify the prime factors of and move any number that appears in a pair of factors from the interior to the exterior of the square root symbol:

$\sqrt{5\cdot5\cdot13}$=5$\sqrt{13}$

What is the distance between the two points and ?

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Remember that you can consider your two points as:

and

From this, remember that the distance formula is:

$\sqrt{(x_1-x_$2)^2$+(y_1-y_$2)^2$}$

Now, for your data, this will look like the following. Be very careful with the negative signs:

or

is only factorable into and ; therefore, your answer is in its final form already.

Find the slope of the equation:

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To determine the slope, we will need the equation in slope-intercept form.

Subtract from both sides.

Divide by negative three on both sides.

$\frac{-3y }{-3}$= $\frac{-2x+4}{-3}$

$y=\frac{2}{3}$x-$\frac{4}{3}$$

The slope is: $\frac{2}{3}$

Find the distance from point to .

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Write the formula to find the distance between two points.

$D = $\sqrt{(x_2-x_$1)^2$+(y_2-y_$1)^2$}$$

Substitute the points into the radical.

$D = $$\sqrt{(-1-3)^2$$+(8-1)^2$}$$

$D = $$\sqrt{(-4)^2$$+(7)^2$}$ = $\sqrt{16+49}$ = $\sqrt{65}$$

The answer is: $\sqrt{65}$

What is the distance between and ?

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Write the distance formula.

$D = $\sqrt{(x_2-x_$1)^2$+(y_2-y_$1)^2$}$$

Substitute the points into the equation.

$D = $$\sqrt{(5-(-2))^2$$+(1-3)^2$}$ = $$\sqrt{7^2$ + $(-2)^2$}$ = $\sqrt{49+4}$$

The answer is: $\sqrt{53}$

A triangle on a coordinate plane has the following vertices: . What is the perimeter of the triangle?

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Since we are asked to find the perimeter of the triangle, we will need to use the distance formula to find the length of each side. Recall the distance formula:

$\text{Distance}$=$\sqrt{(y_2-y_$1)^2$+(x_2-x_$1)^2$}$

Start by finding the distance between the points $(5, 5)$\text{ and }$(10, 2)$ :

Next, find the distance between $(10, 2)$\text{ and }$(8, -2)$ .

Then, find the distance between $(5, 5)$\text{ and }$(8, -2)$ .

Finally, add up the lengths of each side to find the perimeter of the triangle.

$$\text{Perimeter}$=$\sqrt{54}$+$\sqrt{20}$+$\sqrt{58}$=17.92$

Use distance formula to find the distance between the following two points.

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Use distance formula to find the distance between the following two points.

Distance formula is as follows:

$d=$\sqrt{(x_1-x_$2)^2$+(y_1-y_$2)^2$}$$

Note that it doesn't matter which point is "1" and which point is "2" just so long as we remain consistent.

So, let's plug and chug.

$d=$\sqrt{925}$=$\sqrt{25}$$\sqrt{37}$=5$\sqrt{37}$$

So, our answer is

$5$\sqrt{37}$$

What is the distance between the points and ?

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Recall the distance formula:

$\text{Distance}$=$\sqrt{(x_1-x_$2)^2$+(y_1-y_$2)^2$}$

Plug in the given points to find the distance between them.

The distance between those points is $\sqrt{181}$ .

Find the distance between the points and .

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Find the distance between the points and .

To find the distance between two points, we will use distance formula (clever name). Distance formula can be thought of as a modified Pythagorean Theorem. What distance formula does is essentially treats our two points as the ends of a hypotenuse on a right triangle, then uses the two side lengths to find the hypotenuse.

Distance formula:

$D=$\sqrt{(x_1-x_$2)^2$+(y_1-y_$2)^2$}$$

Pythagorean Theorem

If the connection isn't clear, don't worry, we can still solve for distance.

$D=$\sqrt{165520}$$

$D=406.84\approx 407$

So our answer is 407

Find the length of the line connecting the following points.

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Find the length of the line connecting the following points.

To find the length of a line, use distance formula.

$d=$\sqrt{(y_1-y_$2)^2$+(x_1-x_$2)^2$}$$

What we are really doing is making a right triangle and using Pythagorean Theorem to find the hypotenuse.

Let's plug in our points and find the distance!

$d=$\sqrt{(5776+6400}$=$\sqrt{12176}$$

$$\sqrt{12176}$\approx 110$

So our answer is 110

A line has slope $\frac{3}{4}$ and -intercept . Give its -intercept.

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The -intercept will be a point for some value . We use the slope formula

,

setting $m = $\frac{3}{4}$, $x_{1}$ =0, $x_{2}$ = a, $y_{1}$ = -7, $y_{2}$ =0$ ,

and solving for :

$\frac{0- (-7)}{a-0}$ = $\frac{3}{4}$

$\frac{ 7}{a}$ = $\frac{3}{4}$

$$\frac{ 7}{a}$ \cdot a = $\frac{3}{4}$ \cdot a$

$7= $\frac{3}{4}$ \cdot a$

$$\frac{4}$ {3}\cdot 7= $\frac{4}$ {3}\cdot $\frac{3}{4}$ \cdot a$

$a = $\frac{28}$ {3} = 9 $\frac{1}{3}$$

The -intercept is $\left ( 9 $\frac{1}{3}$, 0 \right )$ .

A line has slope $\frac{3}{4}$ and -intercept . Give its -intercept.

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The -intercept will be the point for some value . We use the slope formula

,

setting $m = $\frac{3}{4}$, $x_{1}$ = 5, $x_{2}$ = 0, $y_{1}$ = 0, $y_{2}$ = b$ ,

and solving for :

$\frac{b-0}{0- 5}$ = $\frac{3}{4}$

$\frac{b }{ - 5}$ = $\frac{3}{4}$

$$\frac{b }{ - 5}$ \cdot (-5) = $\frac{3}{4}$ \cdot (-5)$

$b= $\frac{-15}{4}$ = -3 $\frac{3}{4}$$

The -intercept is $\left (0, -3 $\frac{3}{4}$ \right )$ .

Find the midpoint of the line segment that connects the following points:

$\small (-5,-3)\ and\ (3,7)$

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Use the midpoint formula:

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

$\small mid=($\frac{-5+3}{2}$,$\frac{-3+7}{2}$)$

$\small mid=($\frac{-2}{2}$,$\frac{4}{2}$)$

$\small mid=(-1,2)$

You are given points and . is the midpoint of $\overline{AD}$ , is the midpoint of $\overline{AC}$ , and is the midpoint of $\overline{BD}$ . Give the coordinates of .

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Repeated application of the midpoint formula, $\left $($\frac{x_{1}$$$+x_{2}$}{2}, $$\frac{y_{1}$$$+y_{2}$}{2} \right)$ , yields the following:

is the point and is the point $\left(1, 9 \right)$ . is the midpoint of $\overline{AD}$ , so has coordinates

$\left($\frac{8+1}{2}$, $\frac{1+9}{2}$ \right)$ , or $\left( 4.5, 5\right)$ .

is the midpoint of $\overline{AC}$ , so has coordinates

$\left($\frac{8+4.5}{2}$, $\frac{1+5}{2}$ \right)$ , or .

is the midpoint of $\overline{BD}$ , so has coordinates

$\left($\frac{6.25+1}{2}$, $\frac{3+9}{2}$ \right )$ , or .