Mattd wrote:
dynocomet wrote:
Figure ABCD is a rectangle with sides of length x centimeters and width y centimeters, and a diagonal of length z centimeters. What is the measure, in centimeters, of the perimeter of ABCD? (The diagram shows line segment AD labeled as "x" and line segment DC labeled as "y")
(1) x - y = 7
(2) z = 13
I read that the answer is C, but cannot understand why. First of all, statement 1 seems like a contradiction -- how can the hypotenuse be SMALLER than either of the two sides?
Thanks.
The answer to the questions shows C. However why can we not state that from statement (2) we are dealing with a 5,12,13 triangle? Hence B would be sufficient to answer the question.
The point is that a right triangle with hypotenuse 13,
doesn't mean that we have (5, 12, 13) right triangle.
If we are told that the lengths of all sides are integers, then yes: the only integer solution for right triangle with hypotenuse 13 would be (5, 12, 13). Or in other words: \(x^2+y^2=13^2\) DOES NOT mean that \(x=5\) and \(y=12\). Certainly this is one of the possibilities but definitely not the only one. In fact \(x^2+y^2=13^2\) has infinitely many solutions for \(x\) and \(y\) and only one of them is \(x=5\) and \(y=12\).
For example: \(x=1\) and \(y=\sqrt{168}\) or \(x=2\) and \(y=\sqrt{165}\)...
So knowing that the diagonal of a rectangle (hypotenuse) equals to one of the Pythagorean triple hypotenuse values is not sufficient to calculate the sides of this rectangle.
Hope it's clear.
if we were told the sides have integral values then B would've been sufficient?
does that mean whenever we are given for eg 5 as a hypotenuse side and told the other sides have integral values, the other sides would be 3 and 4?
or given 25 as a hypotenuse value, then the other sides (if having integral values) would def be 15 and 20?