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Difference between revisions of "1999 CEMC Pascal Problems/Problem 9"

(Created page with "==Problem== In the diagram, each small square is <math>1</math> cm by <math>1</math> cm. The area of the shaded region, in square centimeters is {{Image needed}} <math> \text{...")
 
 
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{{Image needed}}
 
{{Image needed}}
 
<math> \text{ (A) }\ 2.75 \qquad\text{ (B) }\ 3 \qquad\text{ (C) }\ 3.25 \qquad\text{ (D) }\ 4.5 \qquad\text{ (E) }\ 6</math>
 
<math> \text{ (A) }\ 2.75 \qquad\text{ (B) }\ 3 \qquad\text{ (C) }\ 3.25 \qquad\text{ (D) }\ 4.5 \qquad\text{ (E) }\ 6</math>
−
==Solution==
+
==Solution 1==
 
To find the [[area]] of the shaded region, we can subtract the unshaded region from the total area.
 
To find the [[area]] of the shaded region, we can subtract the unshaded region from the total area.
  
Line 25: Line 25:
  
 
Subtracting this from the total area, we get that the unshaded area is <math>9 - 6 = \boxed {\textbf {(B) } 3}</math> square cm.
 
Subtracting this from the total area, we get that the unshaded area is <math>9 - 6 = \boxed {\textbf {(B) } 3}</math> square cm.
 +
 +
~anabel.disher
 +
==Solution 2==
 +
We can see that the height of the shaded triangle is <math>3</math> cm, and its base is <math>2</math> cm.
 +
{{Image needed}}
 +
We can now use the formula for the area of a triangle to get
 +
 +
<math>\frac{3 \times 2}{2} = \boxed {\textbf {(B) } 3}</math> square cm
  
 
~anabel.disher
 
~anabel.disher

Latest revision as of 19:11, 29 June 2025

Problem

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

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

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

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.

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

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.

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

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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