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
