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Use the enlargement transformation to explain similarity and develop the conditions for triangles to be similar (ACMMG220)
Source: Australian Curriculum, Assessment and Reporting Authority (ACARA)
Similar triangles
Two figures are congruent when they are similar with similarity factor 1. Similarity is thus a generalisation of congruence, and we would expect the theory of similarity to proceed along similar lines to the theory of congruence that we have already developed. As with congruence, discussions involving the similarity of straight-sided figures can be reduced to discussions of similar triangles.
Triangles have three vertex angles and three side lengths. If the vertices of two triangles can be matched up so that matching angles are equal and matching sides are in a constant ratio, then the two triangles are similar.
This can be demonstrated with the two triangles above, where matching angles are equal and matching sides are in the ratio 1 : k. We can enlarge \(\triangle ABC\) on the left by an enlargement with centre A and enlargement factor k to produce \(\triangle A^\prime B^\prime C^\prime,\) which is congruent to \(\triangle PQR\) on the right. This shows that \(\triangle ABC\) is similar to \(\triangle PQR\).
As with congruent triangles, however, we do not need to check all six measurements to be sure that the two triangles are similar. There are four standard tests for two triangles to be similar, corresponding to four of the standard tests for congruence.
| ASA | AAA | SSS | SAS | RHS | |
|---|---|---|---|---|---|
| Congruent | Two – hence three corresponding angles are equal and one pair of matching sides has the same length | Matching sides are equal in length | Two pairs of matching sides and the included angle are equal | Hypotenuses and one other pair of matching sides are equal in a right-angled triangle | |
| Similar | Matching angles are equal | Matching sides have lengths in the same ratio for each pair of sides | Two pairs of matching sides are in the same ratio and the included angle is equal | Hypotenuses and one pair of matching sides are in the same ratio in a right- angled triangle |
We can develop each similarity test from the corresponding congruence test. The proof for AAA is given in the following pages.
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This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations |
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