# SSAT Middle Level Math : Geometry

## Example Questions

### Example Question #1 : How To Find A Triangle On A Coordinate Plane

The isosceles triangle shown above has a perimeter of 22 and base of 6. Find the lengths of the left and right sides, respectively. Assume no other side has a length of 6.

Explanation:

With a perimeter of 22 and base of 6, this means the other two sides must add up to:

Because an isosceles triangle must have two sides equal in length and we know from the problem that no other side equals 6, the two remaining sides must be equal to each other.

Thus, to be equal to each other and also add up to 16, each side must be 8 units in length.

### Example Question #4 : How To Find A Triangle On A Coordinate Plane

Given the above triangle has a base of 5 and height of 6, what is the perimeter of the triangle?

Explanation:

First, use the Pythagorean Theorem to find the length of the hypotenuse:

where  and  are 5 and 6, respectively, and  is the hypotenuse.

Thus,

Finally, the perimeter is the sum of the sides of the triangle or:

### Example Question #1 : How To Find A Triangle On A Coordinate Plane

Given triangle , where  is at point  and  is at point , find the area.

Explanation:

To find the area of this triangle, we first need to determine the length of sides AB and BC. First, point B shares the same x-coordinate as point A and the same y-coordinate as point C. Thus, B must be located at point (-2,-2).

The length of side AB must then be:

and the length of side BC:

Using the area formula,

we can find the area using the base (side BC) and height (side AB):

### Example Question #6 : How To Find A Triangle On A Coordinate Plane

Given triangle , where side  and side , find the perimeter.

Explanation:

Use the Pythagorean Theorem to find the length of side AC:

Then, the perimeter is simply the sum of all three sides:

### Example Question #7 : How To Find A Triangle On A Coordinate Plane

The above triangle has base 6 and height 4. Find the perimeter.

Explanation:

Because the y-axis bisects the base, we can divide the triangle into two, equal right triangles. The base of the right triangle is thus half that of the larger triangle, or 3. The height is still 4. To find the hypotenuse, use the Pythagorean Theorem:

Thus, we now know the base as given in the problem and each of the other two sides (which are also the hypotenuses of the right triangles).

Therefore, the perimeter is:

### Example Question #1 : How To Find A Triangle On A Coordinate Plane

Given a height of  units and base of  units, find the area of the triangle shown above.

Explanation:

The area of any triangle is calculated by the formula:

Thus, the area of this triangle is:

### Example Question #1 : How To Find A Triangle On A Coordinate Plane

Given the above triangle has a base of  and hypotenuse of , find the height of the triangle.

Explanation:

Use the Pythagorean Theorem,

where  is the hypotenuse,  is the base, and  is the height.

Rearranging to solve for the height, , yields:

### Example Question #201 : Ssat Middle Level Quantitative (Math)

Which of the following points would be located in Quadrant III?

Explanation:

By definition, a point on the coordinate plane that is in Quadrant III must have both a negative  coordinate and a negative  coordinate. The only answer choice that satisfies both of these conditions is .

### Example Question #1 : Graphing Points

In which quadrant or on what axis will you find the point ?

The -axis

Explanation:

The point  has a negative  coordinate and a positive  coordinate. By definition, any point on a coordinate plane with these characteristics is located in Quadrant II.

### Example Question #2 : Graphing Points

In which quadrant or on which axis will you find the point ?

The -axis

The -axis