In `triangleABC` measure of angle B is `90^0.` If cosA = `8/17`, and AB = 4cm, then what is the length (in cm) of side BC?

Options

A. 8.5
B. 7.5
C. 5
D. 6

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the triangle and given values
We have a right triangle ABC where angle B is 90 degrees. We are given that cos A = 8/17 and AB = 4 cm.

### Step 2: Use the cosine definition
From the definition of cosine in a right triangle:
\[ \cos A = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \]
In triangle ABC, the adjacent side to angle A is AB, and the hypotenuse is AC. Thus, we can write:
\[ \cos A = \frac{AB}{AC} \]
Substituting the known values:
\[ \frac{8}{17} = \frac{4}{AC} \]

### Step 3: Solve for AC
Cross-multiplying gives:
\[ 8 \cdot AC = 4 \cdot 17 \]
\[ 8 \cdot AC = 68 \]
Now, divide both sides by 8:
\[ AC = \frac{68}{8} = 8.5 \text{ cm} \]

### Step 4: Use Pythagorean theorem to find BC
In a right triangle, according to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
We can rearrange this to find BC:
\[ BC^2 = AC^2 - AB^2 \]
Substituting the values we found:
\[ BC^2 = (8.5)^2 - (4)^2 \]
Calculating the squares:
\[ BC^2 = 72.25 - 16 \]
\[ BC^2 = 56.25 \]

### Step 5: Take the square root to find BC
Taking the square root of both sides:
\[ BC = \sqrt{56.25} = 7.5 \text{ cm} \]

### Final Answer
The length of side BC is **7.5 cm**.

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