HSPT Math - How to do coordinate geometry

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

Explanation

We can use the slope formula:

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

A point is shifted $\textup{3 units}$ to the left and $\textup{5 units}$ down. What is the new point?

Explanation

A point is defined as . If the point is shifted left or right, the value will be added or subtracted. If the point is shifted up or down, then the value will be added or subtracted.

3 units to the left is:

5 units down is:

The new point is:

A line with slope passes through the origin and the point . Evaluate .

$N = -\frac{1}{4}$

Explanation

The origin is . In the following slope formula, set

$x_{2} = 16, y_{2}= N, x_{1} = y_{1}= 0, m = -4$ ,

and solve for :

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

$-4 = \frac{N-0}{16-0}$

$-4 = \frac{N}{16}$

$-4\cdot 16 = \frac{N}{16} \cdot 16$

A deer walks in a straight line for 8 hours. At the end of its journey, the deer is 30 miles north and 40 miles east of where it began. What was the average speed of the deer?

miles per hour

Explanation

To find the speed of the deer, you must have the distance traveled and the time.

The distance is found using the Pythagorean Theorem:

$C = {\sqrt{A^2+ B^2}}$

$C = \sqrt{30^2 + 40^2}$

$C = \sqrt{2500} = 50$

The answer must be in miles per hour so the total miles are divided by the hours to get the final answer:

$\frac{miles}{hour} = \frac{50}{8} = 6.25 mph$

The lines of which two of the following equations are perpendicular to each other?

(I) $y= \frac{6}{11} x$

(II) $y= - \frac{6}{11} x$

(III) $y= - \frac{11}{6} x$

I and III

I and II

II and III

No two of the equations given are represented by lines that are perpendicular to each other.

Explanation

All three equations are in slope-intercept form , so the slope of each line is the coefficient of :

(I) $y= \frac{6}{11} x$ - the slope is $\frac{6}{11}$

(II) $y= - \frac{6}{11} x$ - the slope is $- \frac{6}{11}$

(III) $y= - \frac{11}{6} x$ - the slope is $- \frac{11}{6}$

Two lines are perpendicular if and only if the product of their slopes is .

The product of the slopes of the lines in I and II is

$\frac{6}{11} \times \left (- \frac{6}{11} \right )= - \frac{6 \times 6 }{11 \times 11} =- \frac{36 }{121} e -1$

The product of the slopes of the lines in II and III is

$- \frac{6}{11} \times \left (- \frac{11} {6}\right )= + \frac{6 \times 11 }{11 \times 6} = \frac{66}{66} = 1 e -1$

The product of the slopes of the lines in I and III is

$\frac{6}{11} \times \left (- \frac{11} {6}\right )= - \frac{6 \times 11 }{11 \times 6} =- \frac{66}{66} = -1$

This makes the lines in I and III perpendicular, and this is the correct choice.

Begin at the origin of the rectangular coordinate plane. Move up three units, left seven units, and down nine units. Give the coordinates of your current location.

Explanation

Since you have moved left seven units, you have moved seven units in a negative horizontal direction, making the -coordinate of your current location .

Since you have moved up three units and down nine units, you have moved six units down - this is six units in a negative vertical direction, making the -coordinate of your current location .

Therefore, the ordered pair for your current location is .

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?

$\frac{1}{3}$

$\frac{3}{18}$

$\frac{1}{18}$

$\frac{5}{9}$

Explanation

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 $\frac{6}{18} =\frac{1}{3}$

Which of the following is a vertex of the square?

Question_12

$\small (-1,2)$

$\small (-2,1)$

$\small (-1,-4)$

$\small (1,-4)$

$\small (2,4)$

Explanation

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:
$\small (-1,2); (-1,4); (1,2); (-1,4)$