In the adjoining figure: `anglePSQ= 90^(@)` , PQ= 10 cm , QS = 6cm and RQ = 9cm . Calculate the length of PR.

Answer

To solve the problem step by step, we will use the Pythagorean theorem in two triangles: triangle PQS and triangle PRS.

### Step 1: Identify the given values
- \( PQ = 10 \, \text{cm} \)
- \( QS = 6 \, \text{cm} \)
- \( RQ = 9 \, \text{cm} \)
- \( \angle PSQ = 90^\circ \)

### Step 2: Calculate PS using the Pythagorean theorem in triangle PQS
In triangle PQS, we can apply the Pythagorean theorem:
\[ PQ^2 = PS^2 + QS^2 \]
Substituting the given values:
\[ 10^2 = PS^2 + 6^2 \]
This simplifies to:
\[ 100 = PS^2 + 36 \]
Now, isolate \( PS^2 \):
\[ PS^2 = 100 - 36 \]
\[ PS^2 = 64 \]
Taking the square root gives:
\[ PS = \sqrt{64} = 8 \, \text{cm} \]

### Step 3: Calculate RS
Next, we need to find \( RS \). Since \( RS = RQ + QS \):
\[ RS = 9 \, \text{cm} + 6 \, \text{cm} = 15 \, \text{cm} \]

### Step 4: Calculate PR using the Pythagorean theorem in triangle PRS
Now we can apply the Pythagorean theorem in triangle PRS:
\[ PR^2 = PS^2 + RS^2 \]
Substituting the values we found:
\[ PR^2 = 8^2 + 15^2 \]
Calculating the squares:
\[ PR^2 = 64 + 225 \]
Adding these values gives:
\[ PR^2 = 289 \]
Taking the square root:
\[ PR = \sqrt{289} = 17 \, \text{cm} \]

### Final Answer
The length of \( PR \) is \( 17 \, \text{cm} \).

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