Prove that the area of the equilateral triangle on the hypotenuse of a right triangle equals the sum of the areas of the equilateral triangles on the other two sides.
Proved.
Step 1: Recall the formula for the area of an equilateral triangle with side length \(x\).
Area = \(\dfrac{\sqrt{3}}{4}x^2\).
Step 2: In a right triangle, let the sides be:
Step 3: Draw equilateral triangles on each side \(a\), \(b\), and \(c\). Their areas will be:
Step 4: From the Pythagoras theorem, we know that for a right triangle:
\(c^2 = a^2 + b^2\).
Step 5: Substitute this value of \(c^2\) into the area on side \(c\):
\(A_c = \dfrac{\sqrt{3}}{4}c^2 = \dfrac{\sqrt{3}}{4}(a^2 + b^2)\).
Step 6: Split the terms:
\(A_c = \dfrac{\sqrt{3}}{4}a^2 + \dfrac{\sqrt{3}}{4}b^2 = A_a + A_b\).
Final Result: The area of the equilateral triangle on the hypotenuse is equal to the sum of the areas of the equilateral triangles on the other two sides.