`PQR `is a triangle such that `PQ = PR. RS `and `QT` are the medians to the sides `PQ and PR` respectively. If the medians RS and GT intersect at right angle, then what is the value of `((PQ)/(QR))^2` ?

Options

A. `3/2`
B. `5/2`
C. `2`
D. None of these

Answer

To solve the problem, let's break it down step by step.

### Step 1: Understand the Triangle and Medians
We have triangle \( PQR \) where \( PQ = PR \). This means triangle \( PQR \) is isosceles with \( PQ \) and \( PR \) as the equal sides. The medians \( RS \) and \( QT \) are drawn to sides \( PQ \) and \( PR \) respectively.

**Hint:** Remember that a median in a triangle connects a vertex to the midpoint of the opposite side.

### Step 2: Identify the Intersection of Medians
The medians \( RS \) and \( QT \) intersect at point \( G \) (the centroid of the triangle). According to the problem, these medians intersect at a right angle.

**Hint:** The intersection of the medians in a triangle is the centroid, which divides each median in a 2:1 ratio.

### Step 3: Use the Property of Medians
When two medians intersect at a right angle, there is a specific relationship between the sides of the triangle. The relationship is given by:
\[ PQ^2 + PR^2 = 5QR^2 \]

**Hint:** This property is specific to triangles where the medians intersect at right angles.

### Step 4: Substitute Equal Sides
Since \( PQ = PR \), we can denote both sides as \( x \). Thus, we can rewrite the equation:
\[ x^2 + x^2 = 5QR^2 \]
This simplifies to:
\[ 2x^2 = 5QR^2 \]

**Hint:** Use substitution to simplify the equation when sides are equal.

### Step 5: Solve for \( QR^2 \)
From the equation \( 2x^2 = 5QR^2 \), we can express \( QR^2 \) in terms of \( x^2 \):
\[ QR^2 = \frac{2x^2}{5} \]

**Hint:** Isolate the variable you need to find in the equation.

### Step 6: Find \( \left(\frac{PQ}{QR}\right)^2 \)
Now we need to find \( \left(\frac{PQ}{QR}\right)^2 \):
\[ \frac{PQ}{QR} = \frac{x}{\sqrt{\frac{2x^2}{5}}} = \frac{x}{\frac{x\sqrt{2}}{\sqrt{5}}} = \frac{\sqrt{5}}{\sqrt{2}} \]
Squaring this gives:
\[ \left(\frac{PQ}{QR}\right)^2 = \frac{5}{2} \]

**Hint:** When simplifying fractions, remember to rationalize the denominator if necessary.

### Final Answer
Thus, the value of \( \left(\frac{PQ}{QR}\right)^2 \) is:
\[ \frac{5}{2} \]