Questions: Triangles ABC and DEF are similar triangles. Solve. Round to the nearest tenth. Find the area of triangle ABC.

Solution Steps

• Triangles $ABC$ and $DEF$ are similar.
• The base $DE$ of triangle $DEF$ is 50 cm.
• The height $DF$ of triangle $DEF$ is 25 cm.
• The base $AB$ of triangle $ABC$ is 20 cm.

The area of a triangle is given by the formula: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$

For triangle $DEF$: $$\text{Area}_{DEF} = \frac{1}{2} \times 50 \, \text{cm} \times 25 \, \text{cm} = \frac{1}{2} \times 1250 \, \text{cm}^2 = 625 \, \text{cm}^2$$

Since the triangles are similar, the ratio of their corresponding sides is the same. The ratio of the bases is: $$\text{Scale factor} = \frac{AB}{DE} = \frac{20 \, \text{cm}}{50 \, \text{cm}} = \frac{2}{5}$$

The area of similar triangles is proportional to the square of the scale factor. Therefore: $$\text{Area}_{ABC} = \left( \frac{2}{5} \right)^2 \times \text{Area}_{DEF} = \left( \frac{4}{25} \right) \times 625 \, \text{cm}^2 = 100 \, \text{cm}^2$$

Final Answer