In two isosceles `DeltaABC` and `DeltaPQR`, the ratio of sides AB : PQ is 1 : 3. Find the ratio of area of `DeltaABC` and area of `DeltaPQR`.

Options

A. A)`1:3`
B. B)`1:9`
C. C)`9:1`
D. D)`3:1`

Answer

To solve the problem of finding the ratio of the areas of two isosceles triangles, ΔABC and ΔPQR, given the ratio of their corresponding sides, we can follow these steps:

### Step 1: Understand the Given Information
We know that the ratio of the sides AB to PQ is given as:
\[ AB : PQ = 1 : 3 \]

### Step 2: Identify the Corresponding Sides
Since both triangles are isosceles, we can also assume that:
- \( AC \) (the other equal side of ΔABC) is also equal to \( AB \)
- \( PR \) (the other equal side of ΔPQR) is equal to \( PQ \)

Thus, we have:
\[ AC : PR = 1 : 3 \]

### Step 3: Establish the Ratio of the Sides
From the information provided, we can denote:
- Let \( AB = 1x \)
- Let \( PQ = 3x \)

### Step 4: Use the Area Ratio Formula
The area of similar triangles is proportional to the square of the ratio of their corresponding sides. Therefore, if the ratio of the sides is \( a : b \), then the ratio of the areas will be:
\[ \text{Area ratio} = \left(\frac{a}{b}\right)^2 \]

### Step 5: Calculate the Area Ratio
Substituting the values we have:
\[ \text{Area ratio} = \left(\frac{1}{3}\right)^2 = \frac{1^2}{3^2} = \frac{1}{9} \]

### Conclusion
Thus, the ratio of the area of ΔABC to the area of ΔPQR is:
\[ \text{Area of } ΔABC : \text{Area of } ΔPQR = 1 : 9 \]

### Final Answer
The ratio of the area of ΔABC to the area of ΔPQR is \( 1 : 9 \).

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