Understanding Similar Triangles and Area Ratios

Similar triangles are triangles that have the same shape but potentially different sizes. This means their corresponding angles are equal, and the ratio of their corresponding sides is constant. This constant ratio is called the scale factor.

A key property relating the areas of similar triangles is the Area Ratio Theorem. This theorem states that if two triangles are similar, then the ratio of the area of the first triangle to the area of the second triangle is equal to the square of the ratio of any pair of corresponding sides.

Mathematically, if $\triangle\text{ABC} \sim \triangle\text{RQP}$, then: $$ \frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{RQP})} = \left(\frac{\text{AB}}{\text{RQ}}\right)^2 = \left(\frac{\text{BC}}{\text{QP}}\right)^2 = \left(\frac{\text{AC}}{\text{RP}}\right)^2 $$

Analyzing the Given Information about Similar Triangles

We are given that $\triangle\text{ABC}$ is similar to $\triangle\text{RQP}$. This similarity statement is crucial as it tells us which vertices and sides correspond to each other:

• Vertex A corresponds to Vertex R
• Vertex B corresponds to Vertex Q
• Vertex C corresponds to Vertex P

Based on these correspondences, the corresponding sides are:

• Side AB corresponds to Side RQ
• Side BC corresponds to Side QP (or PQ)
• Side AC corresponds to Side RP (or PR)

We are given the lengths of the sides of $\triangle\text{ABC}$:

• AB = 4 cm
• BC = 6 cm
• AC = 5 cm

We are also given the ratio of the areas of $\triangle\text{ABC}$ and $\triangle\text{PQR}$. Note that $\text{ar}(\triangle\text{PQR})$ is the same as $\text{ar}(\triangle\text{RQP})$.

• ar($\triangle\text{ABC}$) : ar($\triangle\text{PQR}$) = 9 : 4

We need to find the length of side PQ. From the correspondence, side PQ in $\triangle\text{RQP}$ corresponds to side BC in $\triangle\text{ABC}$.

Applying the Area Ratio Theorem to Find PQ

Using the Area Ratio Theorem for the similar triangles $\triangle\text{ABC}$ and $\triangle\text{RQP}$, we have: $$ \frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{RQP})} = \left(\frac{\text{BC}}{\text{QP}}\right)^2 $$ Substitute the given area ratio and the length of BC: $$ \frac{9}{4} = \left(\frac{6}{\text{PQ}}\right)^2 $$

To solve for PQ, take the square root of both sides of the equation: $$ \sqrt{\frac{9}{4}} = \sqrt{\left(\frac{6}{\text{PQ}}\right)^2} $$ $$ \frac{3}{2} = \frac{6}{\text{PQ}} $$

Now, we can cross-multiply or rearrange the equation to find PQ: $$ 3 \times \text{PQ} = 2 \times 6 $$ $$ 3 \times \text{PQ} = 12 $$ $$ \text{PQ} = \frac{12}{3} $$ $$ \text{PQ} = 4 \text{ cm} $$

Conclusion

Using the property that the ratio of areas of similar triangles is equal to the square of the ratio of their corresponding sides, we found the length of PQ. Given the area ratio of 9:4 and the corresponding side BC = 6 cm, the length of PQ is 4 cm.

Revision Table: Key Concepts for Similar Triangles

Additional Information: Properties of Similar Triangles

Beyond the area ratio, similar triangles have several other important properties that are useful in geometry problems:

• Perimeter Ratio: The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides (the scale factor). If the side ratio is $k$, the perimeter ratio is also $k$.
• Altitude Ratio: The ratio of the corresponding altitudes of two similar triangles is equal to the ratio of their corresponding sides.
• Median Ratio: The ratio of the corresponding medians of two similar triangles is equal to the ratio of their corresponding sides.
• Angle Bisector Ratio: The ratio of the corresponding angle bisectors of two similar triangles is equal to the ratio of their corresponding sides.

These properties highlight how linear measurements in similar figures scale by the same factor, while area measurements scale by the square of that factor.