M16-22

Substitute b = (a + 2)/3 and d = a + 2 into a^2 + b^2 = d^2 to get a^2 + ((a + 2)/3)^2 = (a + 2)^2 and simplify.
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\(d=a+2\) and since \(d=3b\), then \(b= \frac{(a+2)}{3}\). Substituting into \(a^2+b^2=d^2\) will give \(a^2−32a−32=0\).
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Official Solution:


If rectangle \(ABCD\) is not a square, what is the perimeter of rectangle \(ABCD\)?

(1) The longer side of the rectangle is 2 meters shorter than its diagonal.

Given: \(a = d - 2\), where \(a\) is the length of the longer side and \(d\) is the length of the diagonal. Not sufficient on its own to get the perimeter.

(2) The ratio of the shorter side of the rectangle to its diagonal is \(\frac{1}{3}\).

Given: \(\frac{b}{d} = \frac{1}{3}\), so \(d=3b\), where \(b\) is the length of the shorter side. Not sufficient on its own to get the perimeter.

(1)+(2) Combining both statements, we get \(a = d - 2\) and \(d = 3b\), which leads to \(a = 3b - 2\). Using the Pythagorean theorem, we have \(a^2 + b^2 = d^2\), which gives \(a^2 - 32a - 32 = 0\). From the quadratic equation, there will be two solutions for \(a\): one positive and another negative. Given the context, the negative solution is invalid. Using the valid positive value for \(a\), we can determine the value of \(b\) and, subsequently, the perimeter. Sufficient.


Answer: C
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Hi guys,

can somebody explain why "a2+b2=d2" is "a2−32a−32=0"?! I cannot see where the 32 is coming from...

Thanks a lot!!!

Hi Bunuel,

Can you please validate my logic. Let breath be "X" then the diagonal is 3X and the Longest side is 3X-2. Now X^2 + (3X-2)^2 = 9(X^2)

Solving this equation we get X^2 - 12X+ 4 =0; using quadratic equation formula for roots ( -b+- (b^2-4ac) )/ 2a . Now we get the two roots as -6 (+/-) 4(sqrt 2) . But in both cases the roots are negative. Where am I making a mistake?

Thanks,
Arun

You should get \(6+4\sqrt{2}\) and \(6-4\sqrt{2}\) as the roots.
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Hi Bunuel

Since you say it's better to solve (when we get an equation like a^2 − 32a −32=0, as we cannot assume that one value will be positive and the other neg.) Would you mind posting a complete resolution to this problem? i.e. including your calculations to get to the positive and negative (invalid) solution.

I am asking this as I think you might be doing the calculus in a much more straightforward way than I am.

Thanks in advance!

Factoring Quadratics: https://www.purplemath.com/modules/factquad.htm
Solving Quadratic Equations: https://www.purplemath.com/modules/solvquad.htm
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I think this is a high-quality question and I agree with explanation.

Longer side of the rectangle \(=x\)

The shorter side of the rectangle \(=y\)

The diagonal of the rectangle \(=d\)

(1) \(x=d-2\); Insufficient

(2) \(\frac{y}{d}=\frac{1}{3}; 3y=d;\) Insufficient

Considering both:
The perimeter \(=2(x+y)\)

The two sides of the rectangle with its diagonal will form a right triangle.

so, \(x^2+y^2=d^2\); from (2) \(d^2=(3y)^2=9y^2\);

\(=(d-2)^2+y^2=9y^2\)

\(=(3y-2)^2+y^2=9y^2\) [Now we will get the value of y and then we can calculate the value x and eventually we will get the perimeter.]

Sufficient.

The answer is \(C\)

is it possible to clarify how did we get to the last equation? a^2−32a−32=0
how did we substitute b ?

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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I was looking for an absolute number in the answer, am i wrong in assuming that -> the question has not asked if we can determine the perimeter but what is the perimeter.

Yes, the question asks what is the perimeter of rectangle ABCD. Here is what the solution says:

(1)+(2) Combining both statements, we get \(a = d - 2\) and \(d = 3b\), which leads to \(a = 3b - 2\). Using the Pythagorean theorem, we have \(a^2 + b^2 = d^2\), which gives \(a^2 - 32a - 32 = 0\). From the quadratic equation, there will be two solutions for \(a\): one positive and another negative. Given the context, the negative solution is invalid. Using the valid positive value for \(a\), we can determine the value of \(b\) and, subsequently, the perimeter. Sufficient.


Answer: C
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