The length of the diagonal of a certain rectangle is 8. If the sum of

Question

The length of the diagonal of a certain rectangle is 8. If the sum of the lengths of the sides is 10, what is the area of the rectangle?

A. 18
B. 36
C. 64
D. 80
E. It cannot be determined.

CounterSniper · Answer

a+ b = 10 
a^2 + b^2 = 100 -2ab
64 = 100-2ab
2ab = 36 , ab=18

NandishSS · Answer

The length of the diagonal of a certain rectangle is 8 i,e d=8

a+b=10 --> (1)

(a+b)^2=100

a^2+b^2+2ab=100

we know that

a^2+b^2 = d^2

64+2ab=100

2ab=36

ab=18

Hence A

PKN · Answer

Given, Diagonal= \sqrt{l^2+w^2}=8  Or,  l^2+w^2=8^2 
Given, sum of the lengths of the sides is 10 Or (l+w)=10

To find, area=l*w=?

(l+w)^2=l^2+w^2+2lw 
Or,  10^2=8^2+2lw 
Or,  l*w=\frac{10^2-8^2}{2} =18

Ans. (A)

JeffTargetTestPrep · Answer

We can create the following equations:

L^2 + W^2 = 8^2

L^2 + W^2 = 64

and

L + W = 10

Squaring both sides of the equation, we have:

(L + W)^2 = 10^2

L^2 + W^2 + 2LW = 100

Substituting, we have:

64 + 2LW = 100

2LW = 36

LW = 18, which is the area of the rectangle.

Answer: A

Valhalla · Answer

Dear experts please help,

Shouldn't the sum of the sides of the rectangle be 2L + 2W, why is it L + W?

Sincerely, Thanks.

shaarang · Answer

I agree the wording is slightly ambiguous but I believe it can be inferred quite clearly that sum of L + W is 10 because it forms a triangle with the diagonal, which is 8, and the sum of two sides needs to be larger than the third side. If you assume the perimeter of the rectangle to be 10 then you can't quite form the triangle with an 8 diagonal.

bryanandre · Answer

I understand the stated solution to the problem, but doesn't the problem make it unable for a diagonal with a length of 8 to exist within a rectangle having an area of 18. Either the two sides are 9 x 2 or 6 x 3. 9 x 2 would make 9 the diagonal of a 9,8,2 triangle and also does not add to 10. 6 x 3 only adds to 9 and would imply the diagonal is √45.

What am I missing here?

Thank you,
Bryan

What am I missing here?

Thank you,
Bryan

Abhishek009 · Answer

l^2 + b^2 = 64  &  l + b = 10

Or,  (l + b)^2 = l^2 + b^2 + 2lb

Or,  (10)^2 = 64 + 2lb

Or,  100 = 64 + 2lb

Or,  36 = 2lb

Or,  lb = 18 , Answer must be (A)

Fdambro294 · Answer

Using the Quadratic Template for "Square of a Sum" will help solve this problem:

Let 1 Side of the Rectangle = X

Let the Other Side of the Rectangle = Y

Q: What is: X * Y = ?

Rule: Rectangles have Adjacent Angles that are 90 Degrees (i.e., every Angle is 90 Degrees) and the Diagonal cuts the Rectangle and creates 2 Right Triangles with the Length and Width as the Legs and the Diagonal as the Hypotenuse

Pythagoras: (X)^2 + (Y)^2 = (8)^2 ---- (eq. 1)

Given: Sum of 2 Sides = 10 -------> (X + Y) = 10 ------ (eq. 2)

(X + Y)^2 = (X)^2 + 2XY + (Y)^2

(X + Y)^2 = (X)^2 + (Y)^2 + 2XY

----substitute (eq. 1) and (eq. 2)----

(10)^2 = (8)^2 + 2XY

100 - 64 = 2XY

XY = 18 = Area of Rectangle

(A)

hiteshdusseja · Answer

Sum Length of all 4 sides will be 10?

CrackverbalGMAT · Answer

CONCEPT 2D Geometry,Simplification,Basic Identities - (Diagonal) ^2= (Length)^2+ (Breath)^2 in a rectangle -(A+B) ^2=A^2+B^2 SOLUTION: Let the sides of the rectangle be A and B. Given, A+B=10 Using the identity mentioned in "Concept " =>100=A^2+B^2 +2AB As explained in "Concept" here , D^2=L^2+B^2 in a rectangle. =>100-2AB=64 =>AB=18 (A) Hope this helps.

keep studying

Devmitra Sen

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