The perimeters of two similar ΔABC and Δ PQR are 48.4 cm and 12.1 cm, respectively. What is the ratio of the areas of Δ ABC and Δ PQR?
16 ∶ 1
Let the two similar triangles be ΔABC and Δ PQR. We are given their perimeters.
- Perimeter of ΔABC = 48.4 cm
- Perimeter of ΔPQR = 12.1 cm
Understanding Similar Triangles and Perimeters
Similar triangles have the same shape, but different sizes. A key property of similar triangles is that the ratio of their corresponding sides is constant. This constant ratio is also equal to the ratio of their perimeters.
Let $s_{ABC}$ represent a side of ΔABC and $s_{PQR}$ represent the corresponding side of ΔPQR. The ratio of their perimeters is:
$$ \frac{\text{Perimeter of } \Delta\text{ABC}}{\text{Perimeter of } \Delta\text{PQR}} = \frac{48.4 \text{ cm}}{12.1 \text{ cm}} $$
Calculating the Ratio of Perimeters
Let's calculate the value of this ratio:
$$ \frac{48.4}{12.1} $$
To make the division easier, we can multiply both numerator and denominator by 10:
$$ \frac{484}{121} $$
We can observe that $121 \times 4 = 484$. So,
$$ \frac{484}{121} = 4 $$
Thus, the ratio of the perimeters of ΔABC to ΔPQR is 4.
Since the ratio of perimeters of similar triangles is equal to the ratio of their corresponding sides, we have:
$$ \frac{s_{ABC}}{s_{PQR}} = 4 $$
Relating Area Ratio to Side Ratio
Another important property of similar triangles is that the ratio of their areas is equal to the square of the ratio of their corresponding sides.
$$ \frac{\text{Area of } \Delta\text{ABC}}{\text{Area of } \Delta\text{PQR}} = \left( \frac{s_{ABC}}{s_{PQR}} \right)^2 $$
Calculating the Ratio of Areas
Using the ratio of sides we found:
$$ \frac{\text{Area of } \Delta\text{ABC}}{\text{Area of } \Delta\text{PQR}} = (4)^2 = 16 $$
So, the ratio of the areas of ΔABC and Δ PQR is 16.
Expressed as a ratio, this is 16 : 1.
Summary of Ratios for Similar Triangles
| Property Ratio | Relationship |
|---|---|
| Ratio of Corresponding Sides | $\frac{a}{p} = \frac{b}{q} = \frac{c}{r} = k$ |
| Ratio of Perimeters | $\frac{\text{Perimeter of } \Delta\text{ABC}}{\text{Perimeter of } \Delta\text{PQR}} = k$ |
| Ratio of Areas | $\frac{\text{Area of } \Delta\text{ABC}}{\text{Area of } \Delta\text{PQR}} = k^2$ |
Step-by-Step Solution Summary
- Identify the given perimeters of the similar triangles ΔABC and Δ PQR.
- Calculate the ratio of the perimeters: $\frac{48.4}{12.1} = 4$.
- Recall that for similar triangles, the ratio of perimeters equals the ratio of corresponding sides. So, the ratio of corresponding sides is 4.
- Recall that for similar triangles, the ratio of areas equals the square of the ratio of corresponding sides.
- Calculate the ratio of areas by squaring the ratio of sides: $4^2 = 16$.
- The ratio of the areas is 16:1.
The ratio of the areas of Δ ABC and Δ PQR is 16 : 1.
Revision Table: Similar Triangle Properties
| Property | Relationship (if ratio of sides is k) |
|---|---|
| Angles | Corresponding angles are equal. |
| Sides | Ratio of corresponding sides is constant (k). |
| Perimeter | Ratio of perimeters is k. |
| Area | Ratio of areas is k<sup>2</sup>. |
| Altitude | Ratio of corresponding altitudes is k. |
| Median | Ratio of corresponding medians is k. |
| Angle Bisector | Ratio of corresponding angle bisectors is k. |
Additional Information: Perimeter and Area Concepts
Perimeter: The perimeter of a polygon is the total distance around its sides. For a triangle with sides a, b, and c, the perimeter is a + b + c. Perimeter is a linear measure.
Area: The area of a polygon is the amount of surface it covers. For a triangle, the area can be calculated using formulas like $\frac{1}{2} \times \text{base} \times \text{height}$. Area is a two-dimensional measure.
When dealing with similar figures, linear measures (like side lengths, perimeters, altitudes, medians) scale by the same factor (the ratio of similarity), while area measures scale by the square of that factor. Volume measures (for 3D similar figures) scale by the cube of the factor.
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