1999 CEMC Pascal Problems/Problem 9

Problem

In the diagram, each small square is $1$ cm by $1$ cm. The area of the shaded region, in square centimeters is

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$\text{ (A) }\ 2.75 \qquad\text{ (B) }\ 3 \qquad\text{ (C) }\ 3.25 \qquad\text{ (D) }\ 4.5 \qquad\text{ (E) }\ 6$

Solution 1

To find the area of the shaded region, we can subtract the unshaded region from the total area.

We can first label the lengths of the legs of the triangles. The unshaded region consists of two right triangles, as all of the angles of a square are right angles.

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The total area is a square with a side length of $3$ cm, meaning that we can find that the area is:

$3^2 = 9$ square cm

We then can find the area of each of the triangles using the formula for the area of a triangle, and add them together:

$\frac{3 \times 3}{2} = \frac{9}{2}$

$\frac{3 \times 1}{2} = \frac{3}{2}$

Adding these, we get:

$\frac{9}{2} + \frac{3}{2} = \frac{9 + 3}{2}$

$=\frac{12}{2} = 6$ square cm

Subtracting this from the total area, we get that the unshaded area is $9 - 6 = \boxed {\textbf {(B) } 3}$ square cm.

~anabel.disher

Solution 2

We can see that the height of the shaded triangle is $3$ cm, and its base is $2$ cm.

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We can now use the formula for the area of a triangle to get

$\frac{3 \times 2}{2} = \boxed {\textbf {(B) } 3}$ square cm

~anabel.disher

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1999 CEMC Pascal Problems/Problem 9

Problem

Solution 1

Solution 2