Question

In a right-angled triangle ABC, ∠B = 90°, BD = 3, DC = 4, and AC = 13. A point D is inside the triangle such as ∠BDC = 90°.

Find the values of  3 - 2 cos ∠BAC + 3 cot ∠BCD

Answer 1

ΔBDC is a right-angled triangle.
∴ BC²
= BD² +DC²
= 3² + 4²
= 9 + 16
= 25
⇒ BC = 5cm
ΔABC is a right-angled triangle.
∴ AB²
= AC² - BC²
= 13² - 5²
= 169 - 25
= 144
⇒  AB = 12cm
 3 - 2 cos ∠BAC + 3 cot ∠BCD

= `3 - 2 xx "AB"/"AC" + 3 xx "DC"/"BD"`

= `3 - 2 xx (12)/(13) + 3 xx (4)/(3)`

= `3 - (24)/(13) + 4`

= `7 - (24)/(13)`

= `(91 - 24)/(13)`

= `(67)/(13)`.