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The length of a side of a rhombus is a two-digit number. If the digits of this number are interchanged the new number formed is one tenth the original number. The sum of these two number is 11. The side and one of the diagonals of the rhombus are in the ratio 5 : 6. What is the area of the rhombus (in square units)?
(A) 49 sq. units
(B) 54 sq. units
(C) 92 sq. units
(D) 96 sq. units
(E) 98 sq. units
(A) 49 sq. units
(B) 54 sq. units
(C) 92 sq. units
(D) 96 sq. units
(E) 98 sq. units
18 Mar 2021, 18:29
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Given: Length of side of rhombus is two digit number. If digits interchanged then \(\frac{1}{10}\) of the original number. Sum of original and new number = 11. Ratio of side and one of diagonal = 5 : 6
Let x be the side length.
y be the new number.
y = \(\frac{x}{10}\) and y + x = 11
=> y = 1, x = 10
So, side length is 10 units.
Let one of the diagonal = k
10 : k = 5 : 6
=> k = 12 units.
Important properties of rhombus:
Diagonals of rhombus bisect each other perpendicularly.
Area of rhombus = \(\frac{diagonal1 * diagonal2}{2}\)
If we take one quarter of rhombus, we get a right angled triangle with hypotenuse 10 units, and one side 6 units (half of diagonal1).
Second side = 8 units (using Pythagoras)
Second diagonal = 16 units
Area = \(\frac{12 * 16}{2}\) = 96 sq units.
So, correct answer is option D.
Let x be the side length.
y be the new number.
y = \(\frac{x}{10}\) and y + x = 11
=> y = 1, x = 10
So, side length is 10 units.
Let one of the diagonal = k
10 : k = 5 : 6
=> k = 12 units.
Important properties of rhombus:
Diagonals of rhombus bisect each other perpendicularly.
Area of rhombus = \(\frac{diagonal1 * diagonal2}{2}\)
If we take one quarter of rhombus, we get a right angled triangle with hypotenuse 10 units, and one side 6 units (half of diagonal1).
Second side = 8 units (using Pythagoras)
Second diagonal = 16 units
Area = \(\frac{12 * 16}{2}\) = 96 sq units.
So, correct answer is option D.
18 Mar 2021, 22:13
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Bunuel
Length of side is 2 digit number and on reversing the digits, you get one tenth of the number.
Say if the side were 21, you would reverse the digits to get 12 which is more than half of 21.
Even if the side were 91, you would reverse the digits to get 19 which is till about a fifth. The only way you would get a tenth is if the number ends with a 0 i.e. if the side is 10 or 20 or 30 etc so that on reversing the digits, you will get 1 or 2 or 3 (one tenth of the number)
Sum of the numbers is 11 which means the side length is 10.
Ratio of side: diagonal = 5 : 6
Since side is 10, diagonal is 12.
A rhombus is a kite shape with all sides equal and diagonals perpendicular bisectors. So you get 4 equal right triangles with side 10 as hypotenuse and one leg as 6 (half the diagonal). This will give us a multiple of 3-4-5 triangle so the other half diagonal will be 8.
Area of triangle = (1/2)*6*8 = 24
Area of rhombus = 4 * 24 = 96
Answer (D)
