In rhombus ABCD, P, Q, R and S are points on AB, BC, CD and DA respectively such that PQRS is a square. The distances of A and B from the respective closest sides of the square are 75 cm and 12 cm. Find the area of PQRS.2,700 cm23,000 cm24,500 cm23,600 cm2
Question
In rhombus ABCD, P, Q, R and S are points on AB, BC, CD and DA respectively such that PQRS is a square. The distances of A and B from the respective closest sides of the square are 75 cm and 12 cm. Find the area of PQRS.2,700 cm23,000 cm24,500 cm23,600 cm2
Solution
To solve this problem, we need to understand the properties of a rhombus and a square.
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In a rhombus, all sides are equal. Let's denote the side of the rhombus (and also the side of the square) as 'a'.
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The distances of A and B from the respective closest sides of the square are 75 cm and 12 cm. This means that the diagonals of the rhombus are 75+12+a and a (since PQRS is a square, the diagonals are equal and are equal to the side of the rhombus).
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The diagonals of a rhombus bisect each other at right angles. So, we can use the Pythagorean theorem to find the length of 'a'.
(75+12+a)^2 = 2a^2 (87+a)^2 = 2a^2 7569 + 174a + a^2 = 2a^2 7569 = a^2 - 174a a^2 - 174a - 7569 = 0
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Solving this quadratic equation, we get the values of 'a' as 189 and -40. Since length cannot be negative, 'a' is 189 cm.
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The area of the square PQRS is a^2 = 189^2 = 35,721 cm^2.
So, none of the options given are correct. The area of the square PQRS is 35,721 cm^2.
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