Question

A triangle has a base of 8 cm and a height of 3 cm. The triangle is dilated buy a scale factor of 3. Calculate and compare the areas of dilated triangle and the original triangle.

Answer 1

To find the areas of the original triangle and the dilated triangle, we can use the formula for the area of a triangle:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

### Step 1: Calculate the area of the original triangle

Given:
- Base = 8 cm
- Height = 3 cm

Calculating the area:

\[
\text{Area}_{\text{original}} = \frac{1}{2} \times 8 \times 3 = \frac{24}{2} = 12 \, \text{cm}^2
\]

### Step 2: Find the dimensions of the dilated triangle

The scale factor of dilation is 3. This means that both the base and height of the triangle will be multiplied by 3.

- New base = \( 8 \, \text{cm} \times 3 = 24 \, \text{cm} \)
- New height = \( 3 \, \text{cm} \times 3 = 9 \, \text{cm} \)

### Step 3: Calculate the area of the dilated triangle

Using the new dimensions:

\[
\text{Area}_{\text{dilated}} = \frac{1}{2} \times 24 \times 9
\]

Calculating the area:

\[
\text{Area}_{\text{dilated}} = \frac{1}{2} \times 216 = 108 \, \text{cm}^2
\]

### Step 4: Compare areas of the dilated triangle and the original triangle

Now we can summarize the areas:

- Area of the original triangle = \( 12 \, \text{cm}^2 \)
- Area of the dilated triangle = \( 108 \, \text{cm}^2 \)

### Ratio of the Areas

The ratio of the area of the dilated triangle to the original triangle is given by the square of the scale factor, which is:

\[
\text{Scale factor} = 3
\]

\[
\text{Area ratio} = \text{Scale factor}^2 = 3^2 = 9
\]

This means the area of the dilated triangle is 9 times larger than that of the original triangle, confirming that:

\[
\text{Area}_{\text{dilated}} = 9 \times \text{Area}_{\text{original}} = 9 \times 12 \, \text{cm}^2 = 108 \, \text{cm}^2
\]

### Conclusion

- Area of the original triangle: \( 12 \, \text{cm}^2 \)
- Area of the dilated triangle: \( 108 \, \text{cm}^2 \)
- The area of the dilated triangle is 9 times the area of the original triangle.