Question

in triangle ABC, the height line drawn from point A divides side BC with lengths 5cm and 12cm. If tan angle A = -17/7, then the area of triangle ABC is..

Answer 1

The size of the angle A is irrelevant in determining the area. All we need is the height and base length. However, we don't know the height, so we do need the apex angle to find the height first!

Let N be the point where the perpendicular from A meets BC. Let BN=12cm and NC=5cm. Let h=AN (height), and the measures of angles BÂN and CÂN be x and y respectively.

tan(x)=BN/AN=12/h and tan(y)=5/h.

tan(x+y)=tan(CÂB)=-17/7=(tan(x)+tan(y))/(1-tan(x)tan(y)) (trig identity).

tan(x)/tan(y)=12/5, so tan(x)=12tan(y)/5=2.4tan(y). Let t=tan(y) and substitute for tan(x):

-17/7=(2.4t+t)/(1-2.4t²)=3.4t/(1-2.4t²),

-1/7=0.2t/(1-2.4t²),

1=1.4t/(2.4t²-1), 2.4t²-1.4t-1=0, which can be written:

12t²-7t-5=(12t+5)(t-1), t=tan(y)=-5/12 or 1=5/h, so tan(x)=-1 or 12/5=12/h. Since h has to be positive, h=5cm, and tan(y)=1 and tan(x)=12/5=2.4.

The base has length 5+12=17cm. Area =½(base)×(height)=½(17)(5)=42.5cm².