Coordinate Geometry - SSAT Middle Level: Quantitative

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Question

Give the slope of the line that passes through and .

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Answer

Use the slope formula, substituting $x_{1}= 4, y_{1}=2,x_{2}= -8, y_{2}=-3$ :

$m =\frac{ y_{2}- y_{1} }{x_{2} - x_{1}}=\frac{ -3-2 }{-8-4} = \frac{ -5 }{-12} = \frac{5 }{12}$

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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?

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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

Give the slope of a line that passes through and .

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Answer

Using the slope formula, substituting $x_{1}=5$ , $x_{2}=2$ , $y_{1}=5$ , and $y_{2}=4$ :

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

$m=\frac{4-5}{2-5}$

Subtract to get:

$m=\frac{-1}{-3}$

Cancel out the negative signs to get:

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

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Question

Give the slope of a line that passes through and .

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Answer

Using the slope formula with $x_{1}=-7$ , $x_{2}=2$ , $y_{1}=4$ , $y_{2}=5$ :

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

$m=\frac{5-4}{2-(-7)}$

Subtract to get:

$m=\frac{1}{9}$

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Question

Give the slope of the line that passes through and .

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Answer

Using the slope formula for

$x_{1}=-5$ , $x_{2}=1$ , $y_{1}=3$ , and $y_{2}=9$ :

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

$m=\frac{9-3}{1-(-5)}$

Combine the negative signs to get:

$m=\frac{9-3}{1+5}$

Subtract and add to get:

$m=\frac{6}{6}$

Reduce to get:

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Question

Give the slope of a line that passes through and .

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Answer

Using the slope formula, where is the slope, $(x_{1},y_{1})=$ , and $(x_{2},y_{2})=$ :

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

$m=\frac{3-7}{2-4}$

$m=\frac{-4}{-2}$

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Question

Find the slope of a line with points and .

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Answer

Using the slope formula, where is the slope, $(x_{1},y_{1})=$ , and $(x_{2},y_{2})=$ :

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

$m=\frac{5-0}{-10-(-1)}$

$m=\frac{5}{-10+1}$

$m=-\frac{5}{9}$

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Question

Find the slope of the line that passes through the points and

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Answer

Using the slope formula, where is the slope, $(x_{1},y_{1})=$ , and $(x_{2},y_{2})=$ :

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

$m=\frac{-4-(-4)}{-5-5}$

$m=\frac{-4+4}{-10}$

$m=\frac{0}{-10}$

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Question

NOTE: Figure NOT drawn to scale.

What is the area of the parallelogram represented by the above figure?

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Answer

The area of a parallelogram is its base multiplied by its height. The diagram below shows both in green:

$A = 9 \cdot5 = 45$

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Question

Find the area of the above parallelogram given a height of 5 units, as shown to scale.

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Answer

Since the diagram is drawn to scale, the base of the parallelogram must be 6 units in length. Using the area formula for a parallelogram:

$A = base \times height = 6 \times 5 = 30$

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Question

Given a base of 12 units and an area of 120 square units, find the height of the paralellogram shown above.

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Answer

The area formula for a parallelogram is:

$A = base \times height$

Thus, to find the height, rearrange:

$height = \frac{A}{base} = \frac{120}{12} = 10$

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Question

The above parallelogram has an area of 30 square units. Which of the following could be the lengths of the base and height of this figure? (Image not drawn to scale.)

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Answer

The formula for the area of a parallelogram is:

$A = base \times height$

Thus, the base and height must equal 30 square units when multiplied together.

3 and 10 are the only combination of base and height which, when multiplied, equal the area of 30 square units.

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Question

Dr. Robinson recently put a rectangular fence around his backyard. The fence has a width of $\small 15$ yards and a length of $\small 8$ yards. If Dr. Robinson paid $\small \small \$ 12.50$ for every yard of fence, how much did the fence cost?

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Answer

To find the cost of the fence, apply the formula $\small P=2(width)+2(length)$ in order to first find the length of the perimeter of the fence.

Then multiply the perimeter by $\small \$12.50$ .

Thus, the solution is:

$\small P=2(15)+2(8)$
$\small P=30+16$
$\small P=46$

$\small 46\times12.5=575$

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Question

In the above figure Dr. Robinson's shed is represented by the red square, and the shed has a perimeter of $\small 12$ yards. Find how much area Dr. Robinson has in his backyard--excluding the area that the shed occupies.

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Answer

To find how much area Dr. Robinson has around his shed in his backyard, first apply the formula: $\small \small P=4S$ , where $\small S$ represents one side of the red shed. Given that $\small \small P=12$
$\small S$ must equal $\small 3$ because $\small 4\times3=12$ .

Now you have enough information to find the area of the red shed.
Apply the formula: $\small A=S^2$
$\small A=3^2=3\times3=9$

Since the rectangular fence has a width of $\small 15$ yards and a length of $\small 8$ yards, apply the formula: $\small Area=width\times length$ in order to find the area of the entire backyard.
Thus,
$\small Area=15\times8=120$

Then simply find the difference between the area of the entire yard and the area of the shed.

Thus the final solution is:
$\small 120-9=111$

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Question

Identify the correct set of coordinates for rectangle $\small ABCD$ .

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Answer

To identify the correct set of coordinate points for rectangle $\small ABCD$ , note that the correct set must have two sets of matching $\small x$ coordinates and two sets of matching $\small y$ coordinates. The only answer choice that meets these specifications is: $\small \small (-11,3),(-11,7),(3,3),(3,7)$

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Question

The rectangle shown above has a width of $\small 14$ and a length of $\small 4$ . Find the area and perimeter of the rectangle.

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Answer

To solve this problem apply the formulas: $\small Area=width\times length$ & $\small Perimeter=2(w)+2(l)$

Thus, the solution is:

$\small A=14\times4=56$
$\small P=2(14)+2(4)=28+8=36$

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Question

Find the area of the above square.

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Answer

In order to find the area of the square above apply the formula: , where the length of one side of the square.

Since ,
the solution is $A=6^2=6\times6=36$

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Question

Find the perimeter of the square shown above.

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Answer

To find the perimeter of this square, apply the formula: , where the length of one side of the square.

Thus, the solution is:

$P=4\times6=24$

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Question

If a square has an area of square units, what is the perimeter?

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Answer

In order to solve this problem, apply the formula , in order to conclude that the length of side must equal for the area to equal .

Once you've found the length of side , apply the formula .

Thus, the solution is $P=4\times7=28$

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Question

A square has an area of square units, what is the perimeter?

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Answer

In order to solve this problem, apply the formula , in order to conclude that the length of side must equal for the area to equal .

Once you've found the length of side , apply the formula

$P=4\times8=32$

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