Answer:
Explanation:
Given ∆PQR with angle bisector QS and sides QP=8 and QR=12, you want to know what percentage the area of ∆QRS is of the area of ∆PQR.
Area formula
The formula for the area of a triangle is ...
A = 1/2bh
where b is the base length and h is the height.
This means triangles of the same height will have areas that are proportional to the lengths of their bases.
Angle bisector
An angle bisector divides the sides of the triangle proportionally. That means ...
SR/QR = SP/QP
SR = QR·SP/QP = 12·6/8
SR = 9 . . . . units
Triangle areas
The area of triangle PQR is ...
A = 1/2bh = 1/2(PR)h = 1/2(6+9)h = (15/2)h
Similarly, the area of triangle QRS is ...
A = 1/2bh = 1/2(RS)h = 1/2(9)h = (9/2)h
Area ratio
The ratio of the triangle areas is ...
AQRS/APQR = ((9/2)h)/((15/2)h) = 9/15 = 3/5 = 60%
The area of triangle QRS is 60% of the area of triangle PQR.
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Additional comment
Here, the problem requires you to work out the base lengths and area formulas for the two triangles. Knowing the angle bisector divides the bases in the same proportion as the sides, we can skip a lot of that and write down the desired percentage as ...
12/(8+12) = 12/20 = 60/100 = 60%