Understanding Similar Triangles and Altitudes

Similar triangles are triangles that have the same shape but may have different sizes. This means that their corresponding angles are equal, and their corresponding sides are proportional. The proportionality of sides is a key characteristic that extends to other corresponding linear measurements within the triangles.

Ratio of Corresponding Sides and Altitudes

For any two similar triangles, there are important relationships between the ratios of their corresponding parts. If the ratio of corresponding sides of two similar triangles is, say, \(a \ratio b\), then the ratio of their corresponding linear elements will also be \(a \ratio b\). These linear elements include:

• Corresponding altitudes
• Corresponding medians
• Corresponding angle bisectors
• Perimeters of the triangles

The ratio of the areas of the triangles, however, is the square of the ratio of their corresponding sides, i.e., \((a \ratio b)^2 = a^2 \ratio b^2\).

Applying the Property to the Question

The question provides the ratio of the corresponding sides of two similar triangles as \(2 \ratio 3\).

According to the property of similar triangles mentioned above, the ratio of their corresponding altitudes is equal to the ratio of their corresponding sides.

Given:

• Ratio of corresponding sides = \(2 \ratio 3\)

Therefore:

• Ratio of corresponding altitudes = Ratio of corresponding sides
• Ratio of corresponding altitudes = \(2 \ratio 3\)

Detailed Explanation

Let the two similar triangles be \(\triangle ABC\) and \(\triangle DEF\). Since they are similar, we have:

\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = \frac{2}{3} \]

Let \(h_1\) be the altitude from vertex A to side BC in \(\triangle ABC\), and \(h_2\) be the corresponding altitude from vertex D to side EF in \(\triangle DEF\).

Consider the triangles formed by the altitudes. Let the altitude from A meet BC at M, and the altitude from D meet EF at N. We now have right-angled triangles \(\triangle ABM\) and \(\triangle DEN\).

In similar triangles \(\triangle ABC\) and \(\triangle DEF\), the corresponding angles are equal. So, \(\angle B = \angle E\).

In \(\triangle ABM\) and \(\triangle DEN\):

• \(\angle AMB = \angle DNE = 90^\circ\) (since AM and DN are altitudes)
• \(\angle B = \angle E\) (corresponding angles of similar triangles \(\triangle ABC\) and \(\triangle DEF\))

By AA similarity criterion, \(\triangle ABM \sim \triangle DEN\).

Since \(\triangle ABM\) and \(\triangle DEN\) are similar, the ratio of their corresponding sides is equal:

\[ \frac{AM}{DN} = \frac{AB}{DE} = \frac{BM}{EN} \]

We know that \(\frac{AB}{DE} = \frac{2}{3}\) (from the given ratio of corresponding sides of \(\triangle ABC\) and \(\triangle DEF\)).

Therefore,

\[ \frac{AM}{DN} = \frac{2}{3} \]

Here, AM represents the altitude \(h_1\) and DN represents the altitude \(h_2\).

So, the ratio of the corresponding altitudes is \(h_1 \ratio h_2 = 2 \ratio 3\).

Summary of Ratios in Similar Triangles

Let the ratio of corresponding sides of two similar triangles be \(a \ratio b\).

Based on this summary and the step-by-step derivation, the ratio of corresponding altitudes is the same as the ratio of corresponding sides.

Conclusion

Given that the ratio of the corresponding sides of the two similar triangles is \(2 \ratio 3\), the ratio of their corresponding altitudes is also \(2 \ratio 3\).

Revision Table: Similar Triangle Ratios

This table helps summarise the key ratios for similar triangles:

Additional Information: Related Concepts

Understanding similar triangles is crucial in geometry. Here are some related concepts:

• Similarity Transformations: Similar triangles can be obtained from one another through a sequence of transformations including translation, rotation, reflection, and dilation (uniform scaling). The ratio of corresponding sides is the scale factor of the dilation.
• Applications of Similar Triangles: Similar triangles are used in various real-world applications, such as surveying, architecture, and photography (pinhole camera).
• Congruent Triangles: Congruent triangles are a special case of similar triangles where the ratio of corresponding sides is \(1 \ratio 1\). They have the same shape and size.

Remember that the ratio of altitudes, medians, and angle bisectors in similar triangles directly follows the ratio of their corresponding sides.