Let's calculate the third vertex for the right triangle step-by-step. We're given two vertices of a right triangle, which are (0, 1) and (0, 6), and the area of the triangle, which is 10 square units.
Step 1: Find the height of the triangle.
Since the given vertices (0, 1) and (0, 6) have the same x-coordinate, they lie on the same vertical line, and thus, the distance between them can be used as the height of the triangle. The height h is simply the difference in the y-coordinates of the two points.
h = |6 - 1| = 5 units
Step 2: Use the area formula for a triangle to find the base.
The area A of a triangle is given by the formula:
A = (1/2) * base * height
We are given the area (A = 10) and the height (h = 5), so we can solve for the base b:
10 = (1/2) * b * 5
20 = b * 5
b = 20 / 5
b = 4 units
Step 3: Find the third vertex.
The third vertex must be such that its distance from either (0, 1) or (0, 6) is equal to the base of 4 units, forming a right angle with the line connecting (0, 1) and (0, 6).
Checking the given options:
A) (5, 6): The second coordinate matches with that of (0, 6), but the distance to (0, 6) along the horizontal axis is 5, which is not equal to the base of 4 units. So, this cannot be the correct third vertex.
B) (6, 6): Again, the second coordinate matches with that of (0, 6), but this point is 6 units away from (0, 6) along the horizontal axis. This does not match our required base, so this cannot be the proper third vertex either.
C) (-5, 1): The second coordinate matches with that of (0, 1), and the distance to (0, 1) along the horizontal axis is 5, which is not equal to the base of 4 units. Thus, this point cannot be the correct third vertex.
D) (-4, 1): The second coordinate matches with that of (0, 1), and the distance to (0, 1) along the horizontal axis is 4, which is the length of the base we calculated. Therefore, the coordinates (-4, 1) are at the correct distance to be the third vertex of the triangle, making this option the correct third vertex.
The correct answer is D) (-4, 1). This point completes the right triangle with the given area of 10 square units and a height of 5 units.
