In `DeltaABC` measure of angle B is `90^(@)`. If cotA = 7/24, and AB = 1.4 cm, then what is the length (in cm) of side AC?

Options

A. 5
B. 4.8
C. 4
D. 5.6

Answer

To find the length of side AC in triangle ABC where angle B is 90 degrees, cot A = 7/24, and AB = 1.4 cm, we can follow these steps:

### Step 1: Understand the cotangent ratio
The cotangent of angle A (cot A) is defined as the ratio of the length of the adjacent side (AB) to the length of the opposite side (BC). Therefore, we can express this as:

\[ \cot A = \frac{AB}{BC} = \frac{7}{24} \]

### Step 2: Substitute the known value of AB
We know that AB = 1.4 cm. We can substitute this value into the cotangent ratio:

\[ \frac{1.4}{BC} = \frac{7}{24} \]

### Step 3: Cross-multiply to find BC
Cross-multiplying gives us:

\[ 1.4 \times 24 = 7 \times BC \]

Calculating the left side:

\[ 33.6 = 7 \times BC \]

### Step 4: Solve for BC
To find BC, divide both sides by 7:

\[ BC = \frac{33.6}{7} \approx 4.8 \text{ cm} \]

### Step 5: Use Pythagorean theorem to find AC
In a right triangle, the Pythagorean theorem states that:

\[ AC^2 = AB^2 + BC^2 \]

Substituting the known values:

\[ AC^2 = (1.4)^2 + (4.8)^2 \]

Calculating the squares:

\[ AC^2 = 1.96 + 23.04 \]

### Step 6: Add the squares
Adding these values gives:

\[ AC^2 = 25 \]

### Step 7: Take the square root to find AC
Now, take the square root of both sides:

\[ AC = \sqrt{25} = 5 \text{ cm} \]

Thus, the length of side AC is **5 cm**.

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