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

Options

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

Answer

To solve the problem step-by-step, we will use the properties of right triangles and the definition of sine.

### Step 1: Understand the triangle and the given values
In triangle ABC, angle B is 90 degrees. We are given:
- \( \sin A = \frac{15}{17} \)
- \( AB = 0.8 \) cm

### Step 2: Use the definition of sine
The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. Therefore, for angle A:
\[ \sin A = \frac{\text{Opposite (BC)}}{\text{Hypotenuse (AC)}} \]
From the given information:
\[ \sin A = \frac{BC}{AC} = \frac{15}{17} \]

### Step 3: Express the hypotenuse in terms of BC
From the sine definition, we can express the hypotenuse (AC) in terms of BC:
\[ AC = \frac{17}{15} \cdot BC \]

### Step 4: Use the Pythagorean theorem
In triangle ABC, we can apply the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
Substituting the known values:
\[ (0.8)^2 + BC^2 = \left(\frac{17}{15} \cdot BC\right)^2 \]

### Step 5: Calculate \(AB^2\)
Calculating \(AB^2\):
\[ AB^2 = 0.8^2 = 0.64 \]

### Step 6: Substitute and simplify
Substituting \(AB^2\) into the equation:
\[ 0.64 + BC^2 = \left(\frac{17}{15} \cdot BC\right)^2 \]
Expanding the right side:
\[ 0.64 + BC^2 = \frac{289}{225} \cdot BC^2 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ 0.64 = \frac{289}{225} \cdot BC^2 - BC^2 \]
Factoring out \(BC^2\):
\[ 0.64 = BC^2 \left(\frac{289}{225} - 1\right) \]
Calculating the fraction:
\[ \frac{289}{225} - 1 = \frac{289 - 225}{225} = \frac{64}{225} \]
Thus, we have:
\[ 0.64 = BC^2 \cdot \frac{64}{225} \]

### Step 8: Solve for \(BC^2\)
Multiplying both sides by \(\frac{225}{64}\):
\[ BC^2 = 0.64 \cdot \frac{225}{64} \]
Calculating:
\[ BC^2 = 0.64 \cdot 3.515625 = 2.25 \]
Taking the square root:
\[ BC = \sqrt{2.25} = 1.5 \text{ cm} \]

### Final Answer
The length of side \(BC\) is \(1.5\) cm.

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