Question

In figure, ∠ABC = 90°, BC = 10 cm, CD = 6 cm, then AD = ______.

Answer 1

In figure, ∠ABC = 90°, BC = 10 cm, CD = 6 cm, then AD = `underlinebb(32/3 cm)`.

Explanation:

Place B(0, 0), C(10, 0), A(0, a). 

Semicircle with diameter BC has equation (x – 5)² + y² = 25. 

Line AC has length L = `sqrt(100 + a^2)`.

The point D lies on AC at distance 6 from C,

So `D = (10, 0) + (6/L)(-10, a)`

= `((10 - 60)/L, (6a)/L)` 

Substitute in the circle equation:

`((5 - 60)/L)^2 + ((6a)/L)^2 = 25` 

With a² = L² – 100 this simplifies to 50L = 3L², so `L = 50/3`. 

Thus AC = `50/3` and AD = AC – CD = `50/3 - 6 = 32/3` cm.