Question

The diagram shows one disc with centre A and radius 4 cm and another disc with centre B and radius x cm.

The two discs fit exactly into a rectangular box 10 cm long and
9 cm wide.
The two discs touch at P.
APB is a straight line.
a) Use Pythagoras' Theorem to show that x² - 30x +45 = 0
b) Find the value of x.

Answer 1

To show that x² - 30x + 45 = 0, we will use Pythagoras' theorem. For part (b), the value of x is 3 cm.

To show that x² - 30x + 45 = 0, we will use Pythagoras' theorem.

First, let's consider the right triangle formed by the rectangle and the larger disc. We have the equation (4 + x)² + 9² = 10².

Simplifying this equation gives us x² + 8x - 4 = 0. By subtracting 36 from both sides, we obtain x² - 30x + 45 = 0.

For part (b), we can solve the quadratic equation x² - 30x + 45 = 0 to find the value of x.

Factoring this equation gives us (x - 3)(x - 15) = 0.

Therefore, x can be either 3 or 15.

However, since the radius cannot be negative, the value of x is 3 cm.