A triangular region is enclosed by the lines $y = \frac{1}{2} x + 3$, $y = -2x + 6$ and $y = 1$. What is the area of the triangular region? Express your answer as a decimal to the nearest hundredth.

Question

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Answer 1

The three equations are:

(1) y=x/2+3 (red line)

(2) y=-2x+6 (blue line)

(3) y=1 (green line)

Draw a picture because it helps a lot (see below). Graph the three lines (y=1 is a horizontal line cutting through the other lines to make a triangle). The apex of the triangle (a right angle) is formed by the intersection of (1) and (2): x/2+3=-2x+6; 5x/2=3, x=6/5=1.2, y=3.6. So the apex is (1.2, 3.6).

Put y=1 in (1) and (2): x=-4 and x=5/2=2.5. The points (-4,1) and (2.5,1) are the endpoints of the base.

From these intersections we can work out the area of the triangle: base=2.5-(-4)=6.5 and height=3.6-1=2.6. The area=(1/2)(base)(height)=(1/2)6.5*2.6=8.45 sq units.