Coordinate Geometry - ISEE Middle Level Math

Card 0 of 13

Question

Give the slope of the line that passes through and .

Answer

Use the slope formula, substituting $x_{1}= 5, y_{1}=3,x_{2}= -1, y_{2}=5$ :

$m =\frac{ y_{2}- y_{1} }{x_{2} - x_{1}}=\frac{ 5-3 }{-1-5} = \frac{ 2 }{-6} =-\frac{1 }{3}$

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Question

What is the slope of the line that passes through and ?

Answer

We can use the slope formula:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{8-4}{8-4}=\frac{4}{4}=1$

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Question

What is the slope of the line that passes through and ?

Answer

We can use the slope formula:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-7-7}{-3-4}=\frac{-14}{-7}=2$

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Question

Billy set up a ramp for his toy cars. He did this by taking a wooden plank and putting one end on top of a brick that was 3 inches high. He then put the other end on top of a box that was 9 inches high. The bricks were 18 inches apart. What is the slope of the plank?

Answer

The value of the slope (m) is rise over run, and can be calculated with the formula below:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The coordinates of the first end of the plank would be (0,3), given that this is the starting point of the plank (so x would be 0), and y would be 3 since the brick is 3 inches tall.

The coordinates of the second end of the plank would be (18,9) since the plank is 18 inches long (so x would be 18) and y would be 9 since the box was 9 inches tall at the other end.

From this information we know that we can assign the following coordinates for the equation:

$\left(x_{1},y_{1}\right) = (0,3)$ and $(x_{2}, y_{2})= (18,9)$

Below is the solution we would get from plugging this information into the equation for slope:

$m = \frac{9-3}{18-0}}$

This reduces to $\frac{6}{18} =\frac{1}{3}$

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Question

Find the slope of the line that passes through coordinates and .

Answer

The formula for slope is:

$m = \frac{y_{2} - y_{1}}{x_{2}-x_{1}}$

In this particular question our values are given as follows.

$y_{2} = -4$

$y_{1} =-3$

$x_{2} = 8$

$x_{1} = -4$

Substituting the above values into the formula for slope we get,

$\frac{-4 - (-3)}{8- (-4)} = \frac{-1}{12}$

$m = -\frac{1}{12}$ .

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Question

What is the slope of the line depicted by the graph?

Answer

Looking at the graph, it is seen that the line passes through the points (-8,-5) and (8,5).

The slope of a line through the points and can be found by setting

$x_{1} = -8,y_{1} = -5, x_{2} = 8, y_{2} = 5$ :

in the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{5-(-5)}{8 - (-8)} = \frac{5+5}{8 + 8} = \frac{10}{16} = \frac{5}{8}$

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Question

Which of the following is a vertex of the square?

Answer

The coordinates of a point are determined by the distance from the origin. The first point in the ordered pair is the number of units to the left or right of the origin. Negative numbers indicate the number of units to the left while positive numbers indicate the number of units to the right. The second number indicates the number of units above or below the origin. Positive numbers indicate the number of units above while negative numbrs indicate the number of units below the origin. The vertices of the square are:
$\small (-1,2); (-1,4); (1,2); (-1,4)$

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Question

Which of the following points will you find on the -axis?

Answer

A point is located on the -axis if and only if it has -coordinate (first coordinate) 0. Of the five choices, only fits that description.

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Question

Which of the following points is on the -axis?

Answer

A point is located on the -axis if and only if it has a -coordinate equal to zero. So the answer is .

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Question

A line segment on the coordinate plane has its endpoints at the points with coordinates $\left ( 3 \frac{1}{3}, - 1 \frac{1}{2}\right )$ and $\left ( 10, 5 \frac{1}{4}\right )$ . Give the coordinates of the midpoint of the segment.

Answer

The -coordinate of the midpoint is half the sum of the -coordinates of the endpoints:

$\frac{1}{2} \times \left ( 3 \frac{1}{3} + 10 \right )$

$= \frac{1}{2} \times 13 \frac{1}{3}$

$= \frac{1}{2} \times \frac{40}{3}$

$= \frac{1}{1} \times \frac{20}{3}$

$= \frac{20}{3}$

$= 6 \frac{1}{3}$

The -coordinate of the midpoint is found similarly:

$\frac{1}{2} \times \left ( - 1 \frac{1}{2} + 5 \frac{1}{4} \right )$

$= \frac{1}{2} \times \left ( 5 \frac{1}{4} - 1 \frac{1}{2} \right )$

$= \frac{1}{2} \times \left ( 5 \frac{1}{4} - 1 \frac{2}{4} \right )$

$= \frac{1}{2} \times \left ( 4 \frac{5}{4} - 1 \frac{2}{4} \right )$

$= \frac{1}{2} \times 3 \frac{3}{4}$

$= \frac{1}{2} \times \frac{15}{4}$

$= \frac{ 15}{8}$

$=1 \frac{7}{8}$

The correct response is $\left (6 \frac{2}{3}, 1 \frac{7}{8}\right )$ .

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Question

A line segment on the coordinate plane has its endpoints at the points with coordinates $\left ( - 300, 200 \right )$ and $\left ( -200, 300 \right )$ . Give the coordinates of the midpoint of the segment.

Answer

The -coordinate of the midpoint can be found by dividing the sum of the -coordinates of the endpoints by 2:

$[ -300 + (-200) ] \div 2 = [ -(300 + 200) ] \div 2 = -500 \div 2 = - (500 \div 2) = -250$

The -coordinate of the midpoint is found similarly:

$(200 + 300) \div 2 = 500 \div 2 = 250$ .

The correct response is .

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Question

A line segment on the coordinate plane has its endpoints at the points with coordinates and . Give the midpoint of the segment.

Answer

The -coordinate of the midpoint can be found by dividing the sum of the -coordinates of the endpoints by 2:

$(7.32 + 10) \div 2 = 17.32 \div 2 = 8.66$

The -coordinate of the midpoints is found similarly:

$(-3.16 + 10) \div 2 =(10 -3.16 ) \div 2 = 6.84 \div 2 = 3.42$

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Question

Angela plotted points A through E on the coordinate plane shown below.

Which point is located at the ordered pair

Answer

Start at point (0, 0) in the bottom, left corner of the the coordinate plane. From there, count right 2 spaces. This is the first number in the given ordered pair. From that point, count up 8 spaces. This is the second number in the given ordered pair.

By moving right 2 and up 8, you should have found Point B.

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