Suppose `DeltaABC` be a right-angled triangle where `angleA= 90^@` and AD is perpendicular to BC. If ar `Delta ABC = 40 cm^2, ar DeltaACD = 10 cm^2` and `bar(AC) = 9 cm`, then the length of BC is

Options

A. 12 cm
B. 18 cm
C. 4 cm
D. 6 cm

Answer

To find the length of BC in triangle ABC, we can follow these steps:

### Step 1: Understand the properties of the triangles
Given that triangle ABC is a right triangle with angle A = 90°, and AD is perpendicular to BC. We also know the areas of triangles ABC and ACD.

### Step 2: Write down the given information
- Area of triangle ABC (ar(ΔABC)) = 40 cm²
- Area of triangle ACD (ar(ΔACD)) = 10 cm²
- Length of AC = 9 cm

### Step 3: Use the similarity of triangles
Since triangles ABC and ACD share angle A and both have a right angle, they are similar by the AA (Angle-Angle) similarity criterion.

### Step 4: Set up the ratio of areas
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Thus, we can write:

\[ \frac{ar(ΔACD)}{ar(ΔABC)} = \left(\frac{AC}{BC}\right)^2 \]

Substituting the known areas:

\[ \frac{10}{40} = \left(\frac{9}{BC}\right)^2 \]

### Step 5: Simplify the equation
Simplifying the left side gives:

\[ \frac{1}{4} = \left(\frac{9}{BC}\right)^2 \]

### Step 6: Cross-multiply to solve for BC
Cross-multiplying gives:

\[ BC^2 = 9^2 \times 4 \]

Calculating \(9^2\):

\[ BC^2 = 81 \times 4 = 324 \]

### Step 7: Take the square root to find BC
Taking the square root of both sides:

\[ BC = \sqrt{324} = 18 \text{ cm} \]

### Final Answer
The length of BC is **18 cm**.

---