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