Question

In the diagram CD is tangent to circle B at point C. If AB = 5 cm, and BD = 13 cm, find AD. Round the answer to the nearest tenth.

Answer 1

`AD \approx 15.6\,cm`

Explanation:

The value of AD is found by using the Pyhagorean Theorem:

`AD = \sqrt{CD^(2)+AC^(2)}`

`AD = \sqrt{CD^(2)+4\cdot AB^(2)}`

`AD^(2) - 4\cdot AB^(2) = CD^(2)`

Besides, BD is measured in terms of the Pythagorean Theorem:

`BD = \sqrt{CD^(2)+AB^(2)}`

`BD^(2) - AB^(2) = CD^(2)`

By Algebra:

`AD^(2)-4\cdot AB^(2) = BD^(2) - AB^(2)`

`AD^(2) = BD^(2)+3\cdot AB^(2)`

`AD = \sqrt{BD^(2)+3\cdot AB^(2)}`

`AD = \sqrt{(13\,cm)^(2)+3\cdot (5\,cm)^(2)}`

`AD \approx 15.6\,cm`

Answer 2

15.6 cm

Explanation:

Given that AB = 5 cm, then BC = 5 cm, and AC = AB+BC = 10 cm

Applying Pythagorean theorem to triangle BCD:

BD² = BC² + CD²

CD = √(BD² - BC²)

CD = √(13² - 5²) = 12 cm

Applying Pythagorean theorem to triangle ACD:

AD² = AC² + CD²

AD = √(AC² + CD²)

AD = √(10² + 12²) = 15.6 cm