A triangle is bound by a semicircle y=sqrt(64-x^2). Write the area A of the triangle as a function of x, A(x).

Question

there is a picture that goes with this, but I cannot add it.

Answer 1

A more direct answer could be given if the picture was provided, but I'll try to give you the steps, assuming that one side lies on the flat side of the semicircle & that the opposite point is on the arc:

• The area for any triangle is A = (1/2) × b × h
• Any side can be the base of the triangle, so if one side of the triangle is on the flat side of the semicircle, use that
• b = the length of the base, so if all points are using variables, then you use the distance equation
  
• b = abs(x₂ - x₁) if the two coordinate points don't change in y-value
• If we use a base on the flat side, and the third point lies on the arc of the semicircle, then the height will always be the dotted line we create the drops down perpendicular to the flat side
• Since the flat side lies on the x-axis the height automatically reduces to the y-value of that third point, which means:
• h = y₃ = sqrt(64 - x₃²)
• Now that we have the mysterious b & h in terms of x, we can write the area in terms of x
• A(x) = (1/2) × abs(x₂ - x₁)  × sqrt(64 - x₃²)

 

Remember this is only for a specific type of triangle that's bound by a semicircle, one that has two of its points on the flat side, and one point on the curved side. Hope this helps!

-Frank

Answer 2

y = √(64 - x²)

y² = 64 - x²

x² + y² = 64

Circle centered at (0, 0) with radius of 8

Base of triangle is 8 and height is 8.

What is the are of the triangle?