Finding the Area of a Similar Triangle

This problem involves similar triangles, $\triangle \text{ABC}$ and $\triangle \text{DEF}$. We are given the lengths of corresponding sides and the area of one triangle, and we need to find the area of the other similar triangle.

Understanding Similar Triangles and Area Ratio

When two triangles are similar, the ratio of their corresponding sides is constant. A very important property related to similar triangles is that the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Mathematically, if $\triangle \text{ABC} \sim \triangle \text{DEF}$, then:

• $\frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{AC}}{\text{DF}}$ (Ratio of corresponding sides is equal)
• $\frac{\text{Area}(\triangle \text{ABC})}{\text{Area}(\triangle \text{DEF})} = \left(\frac{\text{AB}}{\text{DE}}\right)^2 = \left(\frac{\text{BC}}{\text{EF}}\right)^2 = \left(\frac{\text{AC}}{\text{DF}}\right)^2$ (Ratio of areas is the square of the ratio of corresponding sides)

Applying the Area Ratio Property

We are given the following information:

• $\triangle \text{ABC} \sim \triangle \text{DEF}$
• Corresponding sides BC = 4 cm and EF = 5 cm
• Area of $\triangle \text{ABC}$ = 80 cm$^2$

We need to find the Area of $\triangle \text{DEF}$.

Using the area ratio property, we can write the relationship between the areas and the given corresponding sides:

$$ \frac{\text{Area}(\triangle \text{ABC})}{\text{Area}(\triangle \text{DEF})} = \left(\frac{\text{BC}}{\text{EF}}\right)^2 $$

Step-by-Step Calculation

Substitute the given values into the formula:

$$ \frac{80 \text{ cm}^2}{\text{Area}(\triangle \text{DEF})} = \left(\frac{4 \text{ cm}}{5 \text{ cm}}\right)^2 $$

Calculate the square of the ratio of the sides:

$$ \left(\frac{4}{5}\right)^2 = \frac{4^2}{5^2} = \frac{16}{25} $$

So, the equation becomes:

$$ \frac{80}{\text{Area}(\triangle \text{DEF})} = \frac{16}{25} $$

To find Area($\triangle \text{DEF}$), we can rearrange the equation. Multiply both sides by Area($\triangle \text{DEF}$) and by 25, and divide both sides by 16:

$$ \text{Area}(\triangle \text{DEF}) = \frac{80 \times 25}{16} $$

Now, perform the calculation:

We can simplify the expression by noticing that 80 is a multiple of 16 (80 = 16 × 5):

$$ \text{Area}(\triangle \text{DEF}) = \frac{(16 \times 5) \times 25}{16} $$

Cancel out the 16 from the numerator and the denominator:

$$ \text{Area}(\triangle \text{DEF}) = 5 \times 25 $$

$$ \text{Area}(\triangle \text{DEF}) = 125 $$

Since the area of $\triangle \text{ABC}$ was in cm$^2$, the area of $\triangle \text{DEF}$ will also be in cm$^2$.

Therefore, the area of triangle DEF is 125 cm$^2$.

Final Answer Determination

Based on our calculation, the area of $\triangle \text{DEF}$ is 125 cm$^2$. Let's check the given options:

• 169 cm$^2$
• 80 cm$^2$
• 144 cm$^2$
• 125 cm$^2$

Our calculated area matches the fourth option.

Revision Table: Similar Triangles Properties

Additional Information: Solving Geometry Problems

When solving geometry problems involving similar figures, it's helpful to:

• Identify the similar figures and their corresponding vertices and sides.
• Write down the given information clearly.
• Recall the relevant properties of similar figures (sides, angles, perimeter, area).
• Set up an equation based on the property that connects the given and unknown quantities.
• Solve the equation step-by-step.
• Include the correct units in the final answer.

Understanding the relationship between side lengths and areas in similar shapes is crucial for many geometry problems.