NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 6: Triangles - Exercise 6.4

Question 18

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:

• Base = \(a\)
• Height = \(b\)
• Hypotenuse = \(c\)

Step 3: Draw equilateral triangles on each side \(a\), \(b\), and \(c\). Their areas will be:

• On side \(a\): \(A_a = \dfrac{\sqrt{3}}{4}a^2\)
• On side \(b\): \(A_b = \dfrac{\sqrt{3}}{4}b^2\)
• On side \(c\): \(A_c = \dfrac{\sqrt{3}}{4}c^2\)

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.