Question

Recall that in a 30 – 60 – 90 triangle, if the shortest leg measures x units, then the longer leg measures xStartRoot 3 EndRoot units and the hypotenuse measures 2x units.

(150StartRoot 3 EndRoot – 75π) ft2
(300 – 75π) ft2
(150StartRoot 3 EndRoot – 25π) ft2
(300 – 25π) ft2

Answer 1

I don’t see the full problem or diagram — only the reminder about 30–60–90 triangles and four answer choices. Can you paste the whole problem or upload the diagram?

If you want to proceed meanwhile, here are the formulas and the steps I’ll use once I have the diagram/text:

1. 30–60–90 side lengths
- shortest leg = x
- longer leg = x√3
- hypotenuse = 2x

2. Area of the triangle
- A_triangle = (1/2)(shortest)(longer) = (1/2) x (x√3) = (x^2√3)/2

3. Area(s) of circular region(s)
- area of a circle = πr^2; for a semicircle, sector, quarter-circle use the appropriate fraction.
- identify the radius r from the diagram (often r is equal to some side or half a side like x, x√3, or 2x/2 = x).

4. Final area
- typically triangle area minus circular area(s): A = (x^2√3)/2 − (sum of relevant πr^2 terms)

Example inference from your numeric choices:
- Triangle area = 150√3 suggests (x^2√3)/2 = 150√3 ⇒ x^2 = 300 ⇒ x = √300 = 10√3.
- The π terms (75π or 25π) correspond to sum of r^2 values being 75 or 25 (i.e., radii like 5 or √75 etc).

Send the full problem or image and I’ll pick the correct choice and show the work step by step.