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 AC?

Options

A. a. 8.5
B. b. 7.5
C. c. 5
D. d. 6

Answer

To solve the problem step by step, we will use the information given in the question about triangle ABC, where angle B is 90 degrees, and we know the cosine of angle A and the length of side AB.

### Step 1: Understand the triangle
In triangle ABC, angle B is 90 degrees. We have:
- AB = 4 cm (the side opposite to angle C)
- cos A = 8/17

### Step 2: Recall the definition of cosine
The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. For angle A:
\[ \cos A = \frac{\text{Adjacent side (AB)}}{\text{Hypotenuse (AC)}} \]

### Step 3: Set up the equation
From the definition of cosine, we can express this as:
\[ \cos A = \frac{AB}{AC} \]
Substituting the known values:
\[ \frac{8}{17} = \frac{4}{AC} \]

### Step 4: Cross-multiply to solve for AC
Cross-multiplying gives us:
\[ 8 \cdot AC = 4 \cdot 17 \]
This simplifies to:
\[ 8 \cdot AC = 68 \]

### Step 5: Solve for AC
Now, divide both sides by 8 to find AC:
\[ AC = \frac{68}{8} = 8.5 \text{ cm} \]

### Final Answer
The length of side AC is **8.5 cm**.

---