DS Geometry questions

[sanaa.rizwan]

what is the area of the rectangular region R?

1.each diagnosis of R has length 5
2.the perimeter of R id 14

In statement 1

length of diagonal of rectangle = hypotenuse of a 30-60-90 triangle
if we know one side of a right triangle we can find the length of other 2 sides, what would be the length and height of the rectangle, hence area can be determined.
so statement 1 is SUFFICIENT

but the answer to this question is C. How is statement 1 is NOT SUFFICIENT

[Anju@Gurome]

length of diagonal of rectangle = hypotenuse of a 30-60-90 triangle

The triangle can very well be 45-45-90 triangle or 20-70-90 triangle or 40-50-90 triangle. In each case the lengths of the sides will be different and the address of the rectangular region will be different too.

what is the area of the rectangular region R?

1.each diagnosis of R has length 5
2.the perimeter of R id 14

Let us assume the lengths of the two sides of the rectangular region is a and b.
Hence, area of the region = ab

Statement 1: a² + b² = 5² = 25
This is not enough to determine the value of ab.

Not sufficient

Statement 2: 2(a + b) = 14 ---> (a + b) = 14
This is not enough to determine the value of ab.

Not sufficient

1 & 2 Together: Now we can determine the value of ab as (a + b)² = a² + b² + 2ab
So, ab = [(a + b)² - a² - b²]/2

Sufficient

The correct answer is C.

[srcc25anu]

From 1 we know diagonal = 5
But it can be a square (Sqaure is a special rectangle with all sides equal) and in that case area will be 25 / 2 or 12.5
but if diagonal = 5 and sides of recatngle are 4 and 3, then area will be 12
hence Insufficient

From 2 we know Perimeter = 14
again if we assume a sqaure, each side = 14/4 or 3.5 and area will be 3.5^2 = 12.25
but if we assume a rectangle with side 3 and 4, area will be 12
Not sufficient

Together: we know diagonal = 5 and sum of L + B = 7 (since P =14)
thus it has to be a 3*4 rectangle as diag^2 = L^2 + B^2
Area = 3 * 4 = 12
Sufficient. Ans C

[sanaa.rizwan]

I understand what you mean, the diagonal make a rectangle into a 2 right triangles but not necessarily a 30-60-90 or 45-45-90

Also can you explain how did you come with the area of the square to be 12.5 given the length of diagonal = 5

Thanks a lot

[Blue_Skies]

what is the area of the rectangular region R?

1.each diagnosis of R has length 5
2.the perimeter of R id 14

What do you need to find area of a rectangle ? L X B .
With a given diagonal we can adjust L x B and make many rectangles. Hence 1 is Out.
Now perimeter=2(L+B)=14 -> L+B=7.
Now if we know the perimeter and the diagonal than we can find area because we know :

(L+B)square = l2+B2+2LB = 49.
from pythogorus theorm we know l2+b2 = 25. So we can subsitute that above and get 2LB which gives us the area. Hence C is tha answer.