In `trianglePQR` measure of angle Q is 90°. If cosecP = 17/15, and PQ = 0.8cm, then what is the length (in cm) of side QR?

Options

A. 1.7
B. 2
C. 2.5
D. 1.5

Answer

To solve the problem, we need to find the length of side QR in triangle PQR, where angle Q is 90°. We are given that cosec P = 17/15 and PQ = 0.8 cm.

### Step-by-Step Solution:

1. **Understanding Cosecant**:
The cosecant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the opposite side. Therefore, if cosec P = 17/15, it implies:
\[ \text{Hypotenuse} = 17k \quad \text{and} \quad \text{Opposite side (QR)} = 15k \]
for some constant \( k \).

2. **Identifying the Sides**:
In triangle PQR:
- PQ = 0.8 cm (adjacent side to angle P)
- QR = opposite side to angle P
- PR = hypotenuse

3. **Using the Pythagorean Theorem**:
According to the Pythagorean theorem:
\[ PR^2 = PQ^2 + QR^2 \]
Substituting the values we have:
\[ (17k)^2 = (0.8)^2 + (15k)^2 \]
This simplifies to:
\[ 289k^2 = 0.64 + 225k^2 \]

4. **Rearranging the Equation**:
Rearranging gives:
\[ 289k^2 - 225k^2 = 0.64 \]
\[ 64k^2 = 0.64 \]

5. **Solving for k**:
Dividing both sides by 64:
\[ k^2 = \frac{0.64}{64} = 0.01 \]
Taking the square root:
\[ k = 0.1 \]

6. **Finding QR**:
Now, substituting \( k \) back to find QR:
\[ QR = 15k = 15 \times 0.1 = 1.5 \text{ cm} \]

### Final Answer:
The length of side QR is **1.5 cm**.