marcusaurelius wrote:
The hypotenuse of a right triangle is 10 cm. What is the perimeter, in centimeters, of the triangle?
(1) The area of the triangle is 25 square centimeters.
(2) The 2 legs of the triangle are of equal length.
Aside: Plugging in numbers might be challenging, because we'd need to find values that satisfy BOTH the given information (hypotenuse = 10) AND the information in the statements.
IMPORTANT: For geometry DS questions, we are typically checking to see whether the statements "lock" a particular angle or length into having just one value. This concept is discussed in much greater detail the video below.
So, for this question, if a statement FORCES our right triangle into having ONE AND ONLY ONE shape and size, then that statement is sufficient. Moreover, we NEED NOT find the actual perimeter of the triangle. We need only recognize that we
could find its perimeter (finding the perimeter will just waste time).
Okay, onto the question....
Target question: What is the perimeter of the right triangle? Given: The hypotenuse of the triangle has length 10 cm. Statement 1: The area of the triangle is 25 square centimeters. Let's let x = length of one leg
Also, let y = length of other leg
So, if the area is 25, we can write (1/2)xy = 25
[since area = (1/2)(base)(height)]Multiply both sides by 2 to get xy = 50
Multiply both sides by 2 again to get
2xy = 100 [you'll soon see why I performed this step]Now let's deal with the given information (hypotenuse has length 10)
The Pythagorean Theorem tells us that x² + y² = 10²
In other words,
x² + y² = 100We now have two equations:
2xy = 100x² + y² = 100Since both equations are set equal to 100, we can write:
2xy =
x² + y²Rearrange this to get
x² - 2xy + y² = 0Factor to get (x - y)(x - y) = 0
This means that x =y, which means that the two legs of our right triangle HAVE EQUAL LENGTH.
So, the two legs of our right triangle have equal length AND the hypotenuse has length 10.
There is only
one such right triangle in the universe, so statement 1 FORCES our right triangle into having ONE AND ONLY ONE shape and size.
This means that statement 1 is SUFFICIENT
Statement 2: The 2 legs of the triangle are of equal length We already covered this scenario in statement 1.
So, statement 2 is also SUFFICIENT
Answer:
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