The figure below shows graphed on the coordinate plane.
Triangle X-Y-Z with a perpendicular line P-Y from vertex Y on line X-Z forming two right triangles X-Y-P and Z-Y-P. Triangle X-Y-Z has its vertices X at (0, 8), Y at (6, 8) and Z at (5, 3).
What is the area of ?
A.
15 square units
B.
square units
C.
30 square units
D.
square units
To find the area of triangle X-Y-Z, we can first find the length of the base X-Z and the height from vertex Y to line X-Z by using the distance formula.
The distance between two points (x1, y1) and (x2, y2) is given by:
√[(x2 - x1)^2 + (y2 - y1)^2]
The length of X-Z is:
√[(5 - 0)^2 + (3 - 8)^2] = √[25 + 25] = √50 = 5√2
The length of the height from Y to X-Z is the y-coordinate of Y minus the y-coordinate of P:
8 - 3 = 5
Now, we can find the area of triangle X-Y-Z by using the formula:
Area = 1/2 * base * height
Area = 1/2 * 5√2 * 5
Area = 25/2 * √2
Area = 12.5√2 square units
Therefore, the answer is:
B. 12.5√2 square units