In `Delta` PQR ` , /_PQR= 90^@`. If `PR = sqrt"" 7 ` and `PQ - RQ = 2` then ` sin R - sin P = ? `

Options

A. ` (2 )/( sqrt(7))`
B. `1`
C. `0`
D. None of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the triangle and given information
We have a right triangle PQR where angle PQR = 90°. The lengths given are:
- PR (the hypotenuse) = √7
- The difference between the lengths of PQ and RQ is given as PQ - RQ = 2.

### Step 2: Assign variables to the sides
Let:
- PQ = x (the length of side PQ)
- RQ = y (the length of side RQ)

From the information provided, we can express the relationship:
\[ x - y = 2 \]
This implies:
\[ x = y + 2 \]

### Step 3: Use the Pythagorean theorem
In a right triangle, the Pythagorean theorem states:
\[ PR^2 = PQ^2 + RQ^2 \]
Substituting the known values:
\[ (\sqrt{7})^2 = x^2 + y^2 \]
This simplifies to:
\[ 7 = x^2 + y^2 \]

### Step 4: Substitute x in terms of y
Now, substitute \( x = y + 2 \) into the Pythagorean equation:
\[ 7 = (y + 2)^2 + y^2 \]
Expanding the equation:
\[ 7 = (y^2 + 4y + 4) + y^2 \]
Combining like terms:
\[ 7 = 2y^2 + 4y + 4 \]
Rearranging gives:
\[ 2y^2 + 4y - 3 = 0 \]

### Step 5: Solve the quadratic equation
Now we can use the quadratic formula to solve for y:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 2, b = 4, c = -3 \):
\[ y = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2} \]
Calculating the discriminant:
\[ y = \frac{-4 \pm \sqrt{16 + 24}}{4} \]
\[ y = \frac{-4 \pm \sqrt{40}}{4} \]
\[ y = \frac{-4 \pm 2\sqrt{10}}{4} \]
\[ y = \frac{-2 \pm \sqrt{10}}{2} \]

### Step 6: Find the values of x and y
We take the positive root since lengths cannot be negative:
\[ y = \frac{-2 + \sqrt{10}}{2} \]
Now substituting back to find x:
\[ x = y + 2 = \frac{-2 + \sqrt{10}}{2} + 2 \]
\[ x = \frac{-2 + \sqrt{10} + 4}{2} = \frac{2 + \sqrt{10}}{2} \]

### Step 7: Calculate sin R and sin P
Now we can find \( \sin R \) and \( \sin P \):
- \( \sin R = \frac{PQ}{PR} = \frac{x}{\sqrt{7}} \)
- \( \sin P = \frac{RQ}{PR} = \frac{y}{\sqrt{7}} \)

Thus,
\[ \sin R - \sin P = \frac{x - y}{\sqrt{7}} \]

### Step 8: Substitute x and y
Now substituting \( x = y + 2 \):
\[ \sin R - \sin P = \frac{(y + 2) - y}{\sqrt{7}} = \frac{2}{\sqrt{7}} \]

### Final Answer
Thus, the value of \( \sin R - \sin P \) is:
\[ \sin R - \sin P = \frac{2}{\sqrt{7}} \]

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