두 점 사이의 거리와 내분점

For each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.

$3y=-2x+6$

For each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.

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

For each of the following exercises, use the graph in the figure below.

Find the distance between the two endpoints using the distance formula. Round to three decimal places.

For each of the following exercises, use the graph in the figure below.

Find the coordinates of the midpoint of the line segment connecting the two points.

For each of the following exercises, use the graph in the figure below.

Find the distance that $(-3,4)$ is from the origin.

For each of the following exercises, use the graph in the figure below.

Find the distance that $(5,2)$ is from the origin. Round to three decimal places.

For each of the following exercises, use the graph in the figure below.

Which point is closer to the origin?

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 1:value button, hit enter. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y-intercept, for each of the following graphs.

${\text{Y}}_{1}=-2x+5$

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 1:value button, hit enter. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y-intercept, for each of the following graphs.

${\text{Y}}_{1}=\frac{3x-8}{4}$

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 1:value button, hit enter. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y-intercept, for each of the following graphs.

${\text{Y}}_{1}=\frac{x+5}{2}$

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the *x-*intercept or the “zero” to the y-value. Use this to find the x-intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

${\text{Y}}_{1}=-8x+6$

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the *x-*intercept or the “zero” to the y-value. Use this to find the x-intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

${\text{Y}}_{1}=4x-7$

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the *x-*intercept or the “zero” to the y-value. Use this to find the x-intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

${\text{Y}}_{1}=\frac{3x+5}{4}$ Round your answer to the nearest thousandth.

Someone drove 10 mi directly east from their home, made a left turn at an intersection, and then traveled 5 mi north to their place of work. If a road was made directly from the home to the place of work, what would its distance be to the nearest tenth of a mile?

If the road was made in the previous exercise, how much shorter would the person’s one-way trip be every day?

Given these four points: $A(1,3)$, $B(-3,5)$, $C(4,7)$, and $D(5,-4)$ find the coordinates of the midpoint of line segments $\overline{\text{AB}}$ and $\overline{\text{CD}}.$

After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.

Given the graph of the rectangle shown and the coordinates of its vertices, prove that the diagonals of the rectangle are of equal length.

In the previous exercise, find the coordinates of the midpoint for each diagonal.

The coordinates on a map for San Francisco are $(53,17)$ and those for Sacramento are $(128,78)$. Note that coordinates represent miles. Find the distance between the cities to the nearest mile.