Similar Triangles

Similar Triangles are sets of triangles that have the same shape but differ in their respective sizes. These triangles are derived from the base triangles multiplied by a scale factor. We can identify if they are similar by taking note of their corresponding sides, which have the same ratio, and their respective angles which have the same # of arcs.

As with Congruent Triangles, similar triangles can differ from one another based on their orientation through turns, flips and/or slides.

When comparing similar triangles, there are few different scenarios to consider regarding the scale factor, \(k\):

• If \(k > 1\), the larger triangle is an enlargement of the smaller triangle
• If \(0 < k < 1\), the smaller triangle is a reduction of the larger triangle
• If \(k = 1\), the triangles are congruent

Example

Determine the ratio for the following pair of triangles along with the missing side lengths for \(\triangle \text{XYZ}\).

In \(\triangle \text{ABC}\) and \(\triangle \text{XYZ}\), we can identify the corresponding lengths:

\(12 = 8\)

\(15 = y\)

\(9 = z\)

In addition to corresponding lengths, we can also identify the ratio, \(k\), and the missing lengths between the 2 triangles. We can first divide the known side length of \(\triangle \text{XYZ}\) to the corresponding side length of \(\triangle \text{ABC}\) to find the ratio:

\(k = \cfrac{8}{12} = \cfrac{2}{3}\)

We can use this ratio to identify the missing side lengths of \(\triangle \text{XYZ}\) using the corresponding side lengths:

\(y = \cfrac{2}{3}(15) = 10\)

\(z = \cfrac{2}{3}(9) = 6\)

Therefore, we can determine \(\boldsymbol{k = \cfrac{2}{3}}\), \(\boldsymbol{y = 10}\), and \(\boldsymbol{z = 6}\).

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There are several different ways to determine whether a set of triangles are similar without having all the information. We need at least 2-3 values to determine if they're similar.

AA (Angle, Angle)

If 2 triangles have 2 of their known angles equal to each other, this means the triangles are similar. It also means that they share a third angle since the angles must add up to \(180°\).

Example

Determine the value of \(x\) for \(\triangle \text{XYZ}\).

We can identify the corresponsing angles of \(\triangle \text{KLM}\) and \(\triangle \text{XYZ}\):

\(\angle L = \angle Z\)

\(\angle \text{K} = \angle \text{X}\)

As we have identified 2 pairs of corresponding angles, we can infer that the last pair of angles are also corresponding to each other.

We can also determine what the missing side length of \(\triangle \text{XYZ}\) is based on the ratio, \(k\):

\(k = \cfrac{7.5}{5} = \cfrac{3}{2}\)

\(x = \cfrac{3}{2}(4)\)

\(x = 6\)

Therefore, we can determine that the missing side length \(\boldsymbol{x = 6}\).

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SSS (Side, Side, Side)

If 2 triangles share the same ratio across all \(3\) pairs of sides, this means the triangles are similar.

Example

Determine if the following triangles are similar to each other.

We can identify the corresponding side lengths of \(\triangle \text{ABC}\) and \(\triangle \text{DEF}\):

\(\text{AB} = \text{ED}\)

\(\text{BC} = \text{DF}\)

\(\text{AC} = \text{EF}\)

Using these corresponding side lengths, we can determine if they share the same ratios, each represented as \(k\):

\(k_1: \cfrac{4}{10} = \cfrac{2}{5}\)

\(k_2: \cfrac{6}{15} = \cfrac{2}{5}\)

\(k_3: \cfrac{7}{17.5} = \cfrac{2}{5}\)

Therefore, we can determine that \(\triangle \text{ABC}\) and \(\triangle \text{DEF}\) are similar as they share the same ratio across all side lengths.

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SAS (Side, Angle, Side)

If 2 triangles have the same ratio between 2 pairs of sides and share the included angle, this means the triangles are similar. We can also use the Law of Cosines to determine if the triangles are similar.

Example

Determine if the following triangles are similar to each other.

We can identify the corresponding side lengths of \(\triangle \text{PQR}\) and \(\triangle \text{STU}\):

\(\text{AB} = \text{ED}\)

\(\text{BC} = \text{DF}\)

\(\text{AC} = \text{EF}\)

As both triangles have a shared angle, we can determine the ratios, \(k_1\) and \(k_2\), of their corresponding angles to see if they match:

\(k_1: \cfrac{84}{56} = \cfrac{3}{2}\)

\(k_2: \cfrac{87}{58} = \cfrac{3}{2}\)

Therefore, we can determine that \(\text{PQR}\) and \(\triangle \text{STU}\) are similar since they share a common angle and have the same ratio across 2 side lengths.

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