In `DeltaPQR` ,measure of angle Q is `90^(@)` .If `cot P=(8)/(5),andPQ=4` cm., what is the length (in cm.) of side PR ?

Options

A. 8.5
B. 7.5
C. 5
D. 4

Answer

To solve the problem step by step, we will follow the given information and apply trigonometric principles and the Pythagorean theorem.

### Step 1: Understand the triangle and given information
We have a right triangle \( \Delta PQR \) with \( \angle Q = 90^\circ \). We are given:
- \( \cot P = \frac{8}{5} \)
- \( PQ = 4 \) cm

### Step 2: Use the definition of cotangent
The cotangent of angle \( P \) is defined as:
\[ \cot P = \frac{\text{Adjacent side}}{\text{Opposite side}} = \frac{PQ}{QR} \]
Substituting the values we have:
\[ \frac{PQ}{QR} = \frac{8}{5} \]
Since \( PQ = 4 \) cm, we can substitute this into the equation:
\[ \frac{4}{QR} = \frac{8}{5} \]

### Step 3: Solve for \( QR \)
Cross-multiplying gives:
\[ 4 \cdot 5 = 8 \cdot QR \]
\[ 20 = 8 \cdot QR \]
Now, divide both sides by 8:
\[ QR = \frac{20}{8} = 2.5 \text{ cm} \]

### Step 4: Apply the Pythagorean theorem
In a right triangle, the Pythagorean theorem states:
\[ PR^2 = PQ^2 + QR^2 \]
Substituting the known values:
\[ PR^2 = 4^2 + (2.5)^2 \]
Calculating the squares:
\[ PR^2 = 16 + 6.25 \]
\[ PR^2 = 22.25 \]

### Step 5: Find the length of side \( PR \)
Taking the square root of both sides:
\[ PR = \sqrt{22.25} \approx 4.71 \text{ cm} \]

### Final Answer
The length of side \( PR \) is approximately \( 4.71 \) cm.

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