The lengths of side AB and side BC of a scalene triangle ABC are 12 cm and 8 cm respectively. The size of angle C is `90^@`. Find the approximate length of side AC.

Options

A. 12
B. 9
C. 14
D. 16

Answer

To find the approximate length of side AC in triangle ABC, where angle C is 90 degrees, we can use the Pythagorean theorem. Here are the steps to solve the problem:

### Step 1: Identify the sides of the triangle
In triangle ABC:
- Side AB (the hypotenuse) = 12 cm
- Side BC (one leg) = 8 cm
- Side AC (the other leg) = ?

### Step 2: Apply the Pythagorean theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (AB) is equal to the sum of the squares of the lengths of the other two sides (AC and BC). This can be expressed as:
\[ AB^2 = AC^2 + BC^2 \]

### Step 3: Substitute the known values into the equation
Substituting the known lengths into the equation:
\[ 12^2 = AC^2 + 8^2 \]

### Step 4: Calculate the squares
Calculating the squares:
- \( 12^2 = 144 \)
- \( 8^2 = 64 \)

So, the equation becomes:
\[ 144 = AC^2 + 64 \]

### Step 5: Rearrange the equation to solve for AC^2
To isolate \( AC^2 \), subtract 64 from both sides:
\[ AC^2 = 144 - 64 \]
\[ AC^2 = 80 \]

### Step 6: Take the square root to find AC
Now, take the square root of both sides to find AC:
\[ AC = \sqrt{80} \]

### Step 7: Simplify the square root
The square root of 80 can be simplified:
\[ \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \]

### Step 8: Approximate the value of AC
To find an approximate value, we can calculate \( \sqrt{5} \) which is approximately 2.236. Thus:
\[ AC \approx 4 \times 2.236 \approx 8.944 \]

Rounding this to the nearest whole number gives us:
\[ AC \approx 9 \text{ cm} \]

### Final Answer
The approximate length of side AC is 9 cm.

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