Question

pls answer with proof quickly​

Answer 1

Refer the attachment for figure

Name the rectangle as ABCD, A being the centre of bigger circle. The centre of smaller circle is named E. Now join the two centres of the circles to get AE. AE is straight line because it passes through the point of contact of two circles.

Now, length of AE = sum of two radius = r₁ + r₂

ED = CD - CE = r₁ - r₂

Now, angle ADE = 90° (Since ABCD is a rectangle)

So apply Pythagoars theorem to get the length of AD

AD² + DE² = AE²

⇒ AD² = AE² - DE²

⇒ AD² = (r₁ + r₂)² - (r₁ - r₂)²

⇒ AD² = r₁² + r₂² + 2r₁r₂ - r₁² - r₂² + 2r₁r₂

⇒ AD² = 4r₁r₂

⇒ AD = 2√(r₁r₂)

Now, area of rectangle = AD × AB

⇒ 243 = 2√(r₁r₂) × r₁

Square both sides

⇒ (243)² = 4(r₁r₂) × r₁²

Now put the value of r₂ as 6 cm

⇒ (243)² = 4(r₁ × 6) × r₁²

⇒ (243)² = 24 ×  r₁³

⇒ 59,049 = 24 × r₁³

⇒  r₁³ = 59,049/24

⇒  r₁³ = 2460.375

⇒  r₁ = ∛2460.375

⇒  r₁ = 13.5