Question

Two vertices of a triangle are located at (0,4) and (0,10). The area of a triangle is 12 units^2.

what are all the possible positions for the third vertex?
explain how you know these vertices create triangles with an area of 12 units^2
How do you know there aren't anymore?
How many right triangles are there? List the coordinates of the third vertex for each of the right triangle
How many isosceles triangles are there? List the coordinates of the third vertex of the isosceles triangle

Answer 1

The third vertex of the triangle can be located at (4, y) or (-4, y), where y is any real number. To form a right triangle it may be at (4,4), (4,10), (-4,4), or (-4,10). For an isosceles triangle, the third vertex could be at (4,7) or (-4,7).

Step-by-step explanation:

To find the possible positions for the third vertex of the triangle, we need to use the formula for the area of the triangle, which is 1/2 * base * height. We have the area (12 units^2), and the distance between the two given vertices is the height of the triangle (10-4=6 units), so we can find the base. The base is 2*area/height=2*12/6=4 units.

So, the third vertex of the triangle must be 4 units to the right or left of the line x=0, creating the base of the triangle. Therefore, it can be located at either (4, y) or (-4, y), where y is any real number.

To form a right triangle, the third vertex would have to be located at (4,4) or (4,10) or (-4,4) or (-4,10). These four points are on the base and level with the existing points, thus forming a 90-degree angle.

To form an isosceles triangle, the third vertex would be directly across from the midpoint of the segment between the two given vertices. The midpoint between (0,4) and (0,10) is (0,7), so the third vertex would either be at (4,7) or (-4,7).

Learn more about Triangle Geometry