PQRS is a quadrilateral with side \( PS=7 \) cm and \( QR=11 \) cm. \( \angle SPQ \) and \( \angle QRS \) are both right angles. If E and F are points on PQ and RS, respectively, and QE is twice SF; \( SF=n \) cm and n is an integer, then what is the value of n such that the area of the quadrilateral EQFS is \( 225~cm^{2} \)?
Show Hint
- 25
- 15
- 18
- 20
The Correct Option is C
Solution and Explanation
We have a quadrilateral PQRS with right angles at P and R. We are given the area of an inner quadrilateral EQFS and need to find the length \( SF = n \). Step 2: Key Approach:
Decompose the area of quadrilateral EQFS using the diagonal SQ. \[ \text{Area}(EQFS) = \text{Area}(\triangle SQE) + \text{Area}(\triangle SQF) \] Step 3: Detailed Explanation:
Given: - \( \angle SPQ = 90^{\circ} \implies PS \perp PQ \). So, the height of any point on PQ (like E) from vertex S is not directly useful, but the height of vertex S from base PQ is \( PS \). Similarly, for \(\triangle SQE\) with base \( QE \) on line \( PQ \), the altitude from \( S \) to \( PQ \) is \( PS \). - \( \angle QRS = 90^{\circ} \implies QR \perp RS \). For \(\triangle SQF\) with base \( SF \) on line \( RS \), the altitude from \( Q \) to \( RS \) is \( QR \). Let's calculate the areas: 1. Area of \(\triangle SQE\): Base = \( QE \). From the problem, \( QE = 2 \times SF = 2n \). Height = \( PS = 7 \) cm (distance from S to line PQ). \[ \text{Area}(\triangle SQE) = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times (2n) \times 7 = 7n \] 2. Area of \(\triangle SQF\): Base = \( SF = n \). Height = \( QR = 11 \) cm (distance from Q to line RS). \[ \text{Area}(\triangle SQF) = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times n \times 11 = 5.5n \] 3. Total Area: \[ \text{Area}(EQFS) = 7n + 5.5n = 12.5n \] Given Area = 225 \( cm^2 \). \[ 12.5n = 225 \] \[ n = \frac{225}{12.5} = \frac{2250}{125} = 18 \] Step 4: Final Answer:
The value of \( n \) is 18.
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Questions Asked in SRCC GBO exam
Read the given passage and answer the questions.
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