We have seen that a dilation creates similar figures by a transformation that is an enlargement or a reduction of the pre-image by a scale factor. The scale factor is the ratio of corresponding sides. Since the ratios are equal, they create proportions.
In summary, a similarity transformation:
Order Matters
We can use the order of the vertices of the triangle to set up proportions according to the ratios of corresponding side lengths.
\(\triangle{ABC}\sim\triangle{RQS}\) (Reads: Triangle \(ABC\) is similar to Triangle \(RQS\))
- preserves angle measures (corresponding angles are congruent)
- enlarges or reduces side lengths by a scale factor (corresponding lengths are proportional)
Order Matters
We can use the order of the vertices of the triangle to set up proportions according to the ratios of corresponding side lengths.
\(\triangle{ABC}\sim\triangle{RQS}\) (Reads: Triangle \(ABC\) is similar to Triangle \(RQS\))
Using the dilation, we know that \(\overline{AB}\) corresponds with \(\overline{RQ}\), \(\overline{BC}\) corresponds with \(\overline{QS}\), and \(\overline{AC}\) corresponds with \(\overline{RS}\).
So we can set up the proportion: \(\Large\frac{AB}{RQ}=\frac{BC}{QS}=\frac{AC}{RS}\)
Example 1:
Given the two similar triangles in the dilation below, determine the value of \(x\).
So we can set up the proportion: \(\Large\frac{AB}{RQ}=\frac{BC}{QS}=\frac{AC}{RS}\)
Example 1:
Given the two similar triangles in the dilation below, determine the value of \(x\).
Example 2:
Given \(GMTRY\sim HOUSE\), determine the value of \(n\).
Given \(GMTRY\sim HOUSE\), determine the value of \(n\).
Example 3:
Given \(\triangle{TOP}\sim\triangle{AZP}\), determine the value of \(z\).
The composition of transformations is a \(180^{\circ}\) rotation with center \(P\) and a dilation with center \(P\). Here are three ways to solve for the value of \(z\).
Given \(\triangle{TOP}\sim\triangle{AZP}\), determine the value of \(z\).
The composition of transformations is a \(180^{\circ}\) rotation with center \(P\) and a dilation with center \(P\). Here are three ways to solve for the value of \(z\).
Quick Check
1) Error Analysis: Describe and correct the error in finding the missing side length for the similar rectangles.
1) Error Analysis: Describe and correct the error in finding the missing side length for the similar rectangles.