Problem Solving

Priyaranjan wrote:Q. If a square mirror has a 20 inch diagonal, what is the appropriate perimeter of the mirror, in inches?

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

Let x = the length of one side of the square
The diagonal creates a right triangle with each leg having length x and the hypotenuse with length 20.
Plug these values into the Pythagorean Theorem to get: x² + x² = 20²
Simplify: 2x² = 400
Divide both sides by 2 to get x² = 200
So, x = √200
Rewrite as x = 10√2

NOTE: For the GMAT, you should memorize approximations for √2, √3 and √5
√2 ≈ 1.4
So, 10√2 ≈ (10)1.4 ≈ 14

If each side of the square has length 14, then the perimeter of the square = (4)(14) = 56

The best approximation is [spoiler]B (60)[/spoiler]

Cheers,
Brent

Priyaranjan wrote:Sir,

I've got the same result, but followed another rule of the Triangles.

As this was a 90-45-45 degree triangle, the ratio of their sides would be x√2x.
As √2x=20,x=20/√2.
Perimeter= 20*4/√2=80/√2≈57.1... (where √2=1.4)
So, the approximate value would be option B)60.

Kindly let me know about the process.

Thanking you

Hi Priyaranjan,

Your solution is perfect. It's quite similar to mine, except you used the rule about the ratios of a 45-45-90 triangle, and I used the Pythagorean Theorem to (essentially) duplicate that ratio.

Cheers,
Brent

Priyaranjan wrote:Q. If a square mirror has a 20 inch diagonal, what is the appropriate perimeter of the mirror, in inches?

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

Solution:

We are given that a square has a 20-inch diagonal, and we must find the perimeter of the square. We first need to determine the length of the side of the square. To determine the length of the side, we note that the diagonal of a square is also the hypotenuse of a 45-45-90 triangle, with length ratio of s : s : s√2, where s is the length of one of the sides. We thus can use this knowledge to see that: diagonal of a square = side√2 = s√2

20 = s√2

s = 20/√2

Notice that there is a square root in the denominator, which is an inappropriate form. The fraction must be re-expressed by the technique called "rationalizing the denominator." We can multiply 20/√2 by √2/√2. This gives us:

s = (20/√2) × (√2/√2)

s = (20√2)/√4

s = (20√2)/2

s = 10√2

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

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

Answer: B