The areas of two similar trangles `triangleABC and trianglePQR` are 36 sq cms and 9 sq cms respectively. If PQ= 4cm then what is the length of AB (in cm)?

Options

A. 16
B. 12
C. 8
D. 6

Answer

To solve the problem step by step, we will use the properties of similar triangles and the relationship between their areas and corresponding side lengths.

### Step 1: Understand the relationship between the areas of similar triangles
The areas of two similar triangles are proportional to the square of the ratio of their corresponding sides. If the areas of triangle ABC and triangle PQR are A1 and A2 respectively, and the corresponding sides are a1 and a2, then:

\[ \frac{A1}{A2} = \left(\frac{a1}{a2}\right)^2 \]

### Step 2: Substitute the known values
From the problem, we know:
- Area of triangle ABC (A1) = 36 sq cm
- Area of triangle PQR (A2) = 9 sq cm
- Length of side PQ (a2) = 4 cm

We need to find the length of side AB (a1).

### Step 3: Set up the equation
Using the relationship from Step 1, we can set up the equation:

\[ \frac{36}{9} = \left(\frac{AB}{4}\right)^2 \]

### Step 4: Simplify the left side
Calculating the left side gives:

\[ \frac{36}{9} = 4 \]

So, we have:

\[ 4 = \left(\frac{AB}{4}\right)^2 \]

### Step 5: Take the square root of both sides
Taking the square root of both sides, we get:

\[ \sqrt{4} = \frac{AB}{4} \]

This simplifies to:

\[ 2 = \frac{AB}{4} \]

### Step 6: Solve for AB
Now, we can solve for AB by multiplying both sides by 4:

\[ AB = 2 \times 4 = 8 \text{ cm} \]

### Conclusion
The length of AB is 8 cm.

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