jmc

Question

The square with vertices (-1, -1), (1, -1), (-1, 1) and (1, 1) is cut by the line y=x2+ 1 into a triangle and a pentagon. What is the number of square units in the area of the pentagon? Express your answer as a decimal to the nearest hundredth.

Accepted Answer

Sketch the square and the line to find that the line intersects the top side and the left side of the square. Substituting

y = 1

and

x = - 1

into the equation for the line, we find that the intersection points are (0,1) and

(- 1, \frac{1}{2})

. The legs of the removed right triangle (shaded in the figure) measure 1 and 1/2 units, so the area of the triangle is

\frac{1}{2} (1) (\frac{1}{2}) = \frac{1}{4}

square units. Since the area of the whole square is

2^{2} = 4

square units, the area of the pentagon is

4 - \frac{1}{4} = 3.75

square units.