infiniti007 wrote:Can anyone elucidate why "B" would not be the correct answer for the problem below?



ABCD is a rectangle with sides of length x centimeters and width y centimeters, and a diagonal of length z centimeters. What is the perimeter, in centimeters, of ABCD?

1.) x - y = 7
2.) z = 13

Since we are dealing with a right triangle, would not statement 2 imply a 5:12:13 Pythagorean triple? Regardless of which side is 5 or 12, we can determine the perimeter?

We know that x² + y² = z²
Statement 2 tells us that z = 13, so we get: x² + y² = (13²)
In other words, we want x² + y² = 169

There are MANY other possible values of x and y that satisfy this condition. Here are two:

Case a: x = 10, y = √69
Notice that 10² + (√69)² = 169
Here, the perimeter = 20 + 2√69

Case b: x = 1, y = √168
Notice that 1² + (√168)² = 169
Here, the perimeter = 2 + 2√168

Since we get a different perimeter each time, statement 2 is NOT sufficient.

Cheers,
Brent

infiniti007 wrote:Brent - thank you that really helps.

As a quick follow-up, does that mean the concept of using and trying to sight Pythagorean triples is helpful only for when we have 2 known sides then? In other words, just having a right triangle and 1 side of a known Pythagorean triple is not enough to deduce the other sides...

Correct (unless we're given additional information like "the length of each side is an INTEGER")

Cheers,
Brent

infiniti007 wrote:


ABCD is a rectangle with sides of length x centimeters and width y centimeters, and a diagonal of length z centimeters. What is the perimeter, in centimeters, of ABCD?

1) x - y = 7
2) z = 13

Target question: What is the perimeter, in centimeters, of ABCD?

Statement 1: x - y = 7
This tells us that x is 7 cm longer than y.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 8 and y = 1, in which case the perimeter = 8 + 8 + 1 + 1 = 18
Case b: x = 9 and y = 2, in which case the perimeter = 9 + 9 + 2 + 2 = 22
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: z = 13
The Pythagorean Theorem tells that x² + y² = z² (since ∆ABD is a RIGHT triangle)
Statement 2 tells us that z = 13, so we get: x² + y² = (13²)
Simplify to get: x² + y² = 169
There are MANY possible values of x and y that satisfy this condition. Here are two:
Case a: x = 10, y = √69. Notice that 10² + (√69)² = 169. Here, the perimeter = 20 + 2√69
Case b: x = 1, y = √168. Notice that 1² + (√168)² = 169. Here, the perimeter = 2 + 2√168
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x - y = 7
Let's use this to rewrite y in terms of x to get: y = x - 7

Statement 2 tells us that z = 13
This means that: x² + y² = (13²)
Simplify to get: x² + y² = 169

Now replace y with x - 7 to get: x² + (x - 7)² = 169
Expand: x² + x² - 14x + 49 = 169
Simplify: 2x² - 14x - 120 = 0
Divide both sides by 2 to get: x² - 7x - 60 = 0
Factor: (x - 12)(x + 5) = 0
Solve to get x = 12 or x = -5
Since x CANNOT be negative, we can be certain that x = 12.
This means that y = 5
And this means the perimeter = 12 + 12 + 5 + 5 = 34
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent