In `Delta`ABC measure of angle B is `90^@`. If sinA = 15/17, and AB = 0.8cm, then what is the length (in cm) of side AC?

Options

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

Answer

To find the length of side AC in triangle ABC, where angle B is 90 degrees, we can follow these steps:

### Step 1: Understand the Triangle and Given Values
We have a right triangle ABC with:
- Angle B = 90 degrees
- sin A = 15/17
- AB = 0.8 cm

### Step 2: Use the Definition of Sine
The sine of angle A in a right triangle is defined as the ratio of the length of the opposite side (BC) to the length of the hypotenuse (AC). Thus, we can write:
\[ \sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{BC}{AC} \]
Given that \(\sin A = \frac{15}{17}\), we can set:
- BC = 15k (where k is a scaling factor)
- AC = 17k

### Step 3: Use the Pythagorean Theorem
According to the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
Substituting the known values:
\[ (0.8)^2 + (15k)^2 = (17k)^2 \]
Calculating \(0.8^2\):
\[ 0.64 + 225k^2 = 289k^2 \]

### Step 4: Rearranging the Equation
Now, rearranging the equation:
\[ 289k^2 - 225k^2 = 0.64 \]
\[ 64k^2 = 0.64 \]
Dividing both sides by 64:
\[ k^2 = \frac{0.64}{64} = 0.01 \]
Taking the square root:
\[ k = 0.1 \]

### Step 5: Find the Length of AC
Now, substituting the value of k back into the expression for AC:
\[ AC = 17k = 17 \times 0.1 = 1.7 \text{ cm} \]

### Conclusion
The length of side AC is **1.7 cm**.

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