How to find a triangle on a coordinate plane - SSAT Middle Level Quantitative

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:

Use the Pythagorean Theorem,

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

Rearranging to solve for the height, , yields:

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):

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

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

Using the coordinate system, one can see the base of the triangle is 6 units in length. Since it is an equilateral triangle, the other two sides must also be 6 units each in length. Therefore the perimeter is:

The area of any triangle is calculated by the formula:

Thus, the area of this triangle is:

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.

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:

Because this is a right triangle, the area formula is simply:

Thus, the solution is:

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:

Use the Pythagorean Theorem,

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

Rearranging to solve for the height, , yields:

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):

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

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

Using the coordinate system, one can see the base of the triangle is 6 units in length. Since it is an equilateral triangle, the other two sides must also be 6 units each in length. Therefore the perimeter is:

The area of any triangle is calculated by the formula:

Thus, the area of this triangle is:

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.

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:

Because this is a right triangle, the area formula is simply:

Thus, the solution is:

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: