Question

1. solve for x. then find the side lengths of the triangle. 2x ft 6x ft 200 ft

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\). Here, \(a = 2x\), \(b = 200\), \(c = 6x\).
So, \((2x)^{2}+200^{2}=(6x)^{2}\).

Step2: Expand the equation

Expand \((2x)^{2}=4x^{2}\), \((6x)^{2}=36x^{2}\). The equation becomes \(4x^{2}+40000 = 36x^{2}\).

Step3: Rearrange the equation

Subtract \(4x^{2}\) from both sides: \(40000=36x^{2}-4x^{2}\).
Simplify the right - hand side: \(40000 = 32x^{2}\).

Step4: Solve for \(x^{2}\)

Divide both sides by 32: \(x^{2}=\frac{40000}{32}=1250\).

Step5: Solve for \(x\)

Take the square root of both sides. Since \(x>0\) (length cannot be negative), \(x=\sqrt{1250}=\sqrt{25\times50}=5\sqrt{50}=25\sqrt{2}\approx35.36\).

Step6: Find the side lengths

• One side: \(2x = 2\times25\sqrt{2}=50\sqrt{2}\approx70.71\) ft.
• Another side: \(6x=6\times25\sqrt{2}=150\sqrt{2}\approx212.13\) ft.

Answer:

\(x = 25\sqrt{2}\approx35.36\), side lengths are \(50\sqrt{2}\approx70.71\) ft, \(200\) ft, \(150\sqrt{2}\approx212.13\) ft.