In a `Delta` PQR, perpendicular Ps from P to QR meets QR at S. If PS : QS : RS = 2 : 4 : 1, then which of the following is correct ?

Options

A. PQR is an equilateral triangle
B. PQR is right angled at P
C. PQR is an isosceles triangle
D. PQ = 3 PR

Answer

To solve the problem step by step, we will analyze the triangle PQR with the given ratios.

### Step 1: Understand the Problem
We have a triangle PQR with a perpendicular PS from point P to line QR, meeting at point S. The segments PS, QS, and RS are in the ratio 2:4:1.

### Step 2: Assign Variables
Let’s assign a common multiple to the segments based on the given ratio:
- PS = 2X
- QS = 4X
- RS = X

### Step 3: Calculate QR
Since QR is composed of QS and RS, we can calculate QR:
- QR = QS + RS = 4X + X = 5X

### Step 4: Draw the Diagram
Draw triangle PQR with point S on line QR such that PS is perpendicular to QR. Label the segments according to the values we assigned:
- PS = 2X
- QS = 4X
- RS = X

### Step 5: Apply Pythagorean Theorem
Now, we will analyze the two right triangles formed:
1. Triangle PSR (right triangle at S):
- PR² = PS² + SR²
- PR² = (2X)² + (X)²
- PR² = 4X² + X² = 5X²
- PR = √(5X²) = √5 * X

2. Triangle PSQ (right triangle at S):
- PQ² = PS² + QS²
- PQ² = (2X)² + (4X)²
- PQ² = 4X² + 16X² = 20X²
- PQ = √(20X²) = √20 * X = 2√5 * X

### Step 6: Use Pythagorean Theorem for the Whole Triangle
Now we will check if the triangle PQR satisfies the Pythagorean theorem:
- QR² = QS² + SR²
- QR² = (4X)² + (X)²
- QR² = 16X² + X² = 17X²
- QR = 5X, so QR² = (5X)² = 25X²

### Step 7: Check the Relationship
We need to check if:
- PQ² + PR² = QR²
- (2√5 * X)² + (√5 * X)² = (5X)²
- 20X² + 5X² = 25X²
- 25X² = 25X² (True)

### Conclusion
Since the Pythagorean theorem holds, triangle PQR is a right triangle at point S.

### Final Answer
The correct option is B: PQR is a right angle triangle at point S.

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