Problem Solving

pzazz12 wrote:If a square mirror has a 20-inch diagonal, what is the approximate perimeter of the mirror, in inches?

A. 40
B. 60
C. 80
D. 100
E. 120

We are given that a square has a 20-inch diagonal, and we must solve for the perimeter of the square. This means we first need to know the length of a side of the square. To determine the length of a side we can use the diagonal formula for a square. We know that:

diagonal of a square = side√2 = s√2

20 = s√2

s = 20/√2

To rationalize the denominator, we multiply 20/√2 by √2/√2. This gives us:

s = 20/√2 × √2/√2 = 20√2/√4

s = 20√2/2 = 10√2

We also should have memorized that the approximate value of √2 is 1.4. Thus one side of the square is approximately 10 x 1.4 = 14 inches.

Finally, the perimeter of the square is approximately 4 x 14 = 56 inches. Since we are asked to approximate, the closest answer is 60.

Alternate Solution:

Since the diagonal is 20 and since the diagonal of a square is √2 times the side of a square, we have

s√2 = 20

s = 20/√2 = 10√2

where s denotes the side of the square. Thus, the perimeter of the square is 4 x 10√2 = 40√2.

Instead of estimating the side of a square, let's compare the square of the perimeter to the possible answers. We see that the square of the perimeter is (40√2)^2 = 40^2 x (√2)^2 = 1600 x 2 = 3200.

Now, let's square the possible perimeters:

A) perimeter = 40, perimeter^2 = 1600

B) perimeter = 60, perimeter^2 = 3600

C) perimeter = 80, perimeter^2 = 6400

At this point, we don't even need to check answer choices D and E because the square of those numbers will be even greater than 6400. The choice whose square is closest to 3200 is B.

Answer: B