When you look at two figures, they must have the same shape (though not necessarily the same size) in order to be similar. The symbol we use to show that two polygons are similar is: \(\sim\)
Are the two figures similar? If not, move the vertices of \(K'I'T'E'\) so that the figures are similar.
Are the two figures similar? If not, move the vertices of \(K'I'T'E'\) so that the figures are similar.
Similar (\(\sim\)) Polygons
- The corresponding sides of pre-image to image are proportional (which means you can find a common scale factor between each pair of corresponding sides),
- The corresponding angles of pre-image to image are congruent.
Both of the above statements must be true in order to prove that polygons are similar. Otherwise, there is not enough information to tell if they are. A composition of transformations including a dilation and possibly isometries (translation, reflection, rotation) can be used to map one polygon onto the other.
Example 1:
Are the following polygons similar? If they are similar, write a similarity statement. If they are not similar, explain why not.
Example 1:
Are the following polygons similar? If they are similar, write a similarity statement. If they are not similar, explain why not.
Solution:
Notice that each of the corresponding angles are congruent. Now let’s check to see if the corresponding sides are congruent by setting up ratios:
\(\dfrac{13.416}{4.472}=3\;\;\;\;\;\) \(\dfrac{15}{5}=3\;\;\;\;\;\) \(\dfrac{16.155}{5.385}=3\;\;\;\;\;\) \(\dfrac{15.297}{5.099}=3\)
Since each of these ratios are the same, the corresponding sides of the polygon are proportional. Thus, \(ABCD \sim A’B’C’D’\) because the corresponding angles are congruent and the corresponding sides are proportional. Notice that the order in which you write your pre-image vertices should match with which vertex in the image in which it corresponds.
Example 2:
Use the measuring tools to determine whether or not these rectangles are similar.
Notice that each of the corresponding angles are congruent. Now let’s check to see if the corresponding sides are congruent by setting up ratios:
\(\dfrac{13.416}{4.472}=3\;\;\;\;\;\) \(\dfrac{15}{5}=3\;\;\;\;\;\) \(\dfrac{16.155}{5.385}=3\;\;\;\;\;\) \(\dfrac{15.297}{5.099}=3\)
Since each of these ratios are the same, the corresponding sides of the polygon are proportional. Thus, \(ABCD \sim A’B’C’D’\) because the corresponding angles are congruent and the corresponding sides are proportional. Notice that the order in which you write your pre-image vertices should match with which vertex in the image in which it corresponds.
Example 2:
Use the measuring tools to determine whether or not these rectangles are similar.
Triangles tell a different story. Since triangles only have three sides, triangles don’t need both statements to be proven similar. Instead, it’s one or the other:
Similar Triangles
- The corresponding sides of pre-image to image are proportional. This is called side side side similarity (\(SSS\sim\)).
- The corresponding angles of pre-image to image are congruent. This is called angle angle similarity (\(AA\sim\)).
Example 3:
Are the following triangles similar? If they are similar, write a similarity statement. If they are not similar, explain why not. Solution: The \(m\angle B = 180^{\circ} - 82^{\circ} - 60^{\circ} = 38^{\circ}\). With \(\angle C\cong\angle E\) and \(\angle B\cong\angle D\) the triangles will be similar. Therefore \(\triangle ABC\cong\triangle FDE\) by \(AA\sim\) |
Example 4:
Are the following triangles similar? If they are similar, write a similarity statement. If they are not similar, explain why not. Solution: The ratio of each of corresponding sides is \(\Large\frac{VC}{TR} = \frac{4}{2} = \normalsize 2\), \(\Large\frac{VE}{TO} = \frac{8}{4} = \normalsize 2\), \(\Large\frac{EC}{OR} = \frac{\sqrt{80}}{\sqrt{20}} = \normalsize\sqrt4 = 2\). Since the corresponding sides are proportional the triangles are similar. Therefore \(\triangle VEC\cong\triangle TOR\) by \(SSS\sim\). |
Since these triangles are similar, we can use a composition of transformations to map \(\triangle TOR\) onto \(\triangle VEC\). One composition that will work is a reflection followed by a dilation.
Example 5:
What must be the coordinates of \(F\) to make the two triangles similar? Give as many solutions as possible, and include an explanation of why the triangles will be similar. Solution: We can find the lengths of each side since \(\triangle ABC\) is on a coordinate plane. To find \(AB\) and \(AC\), we can use the diatance formula. \(AB = \sqrt{(3 - 1)^2 + (5 - -3)^2} = \sqrt{(2)^2 + (8)^2}\) \(= \sqrt{4 + 64} = \sqrt{68}\) \(AC = \sqrt{(3 - 5)^2 + (5 - -3)^2} = \sqrt{(-2)^2 + (8)^2}\) \(= \sqrt{4 + 64} = \sqrt{68}\) \(BC = 4\) Notice that \(\triangle ABC\) is isosceles, which means a triangle that is similar to it will also be isosceles. Using the coordinates for \(D\) and \(E\): \(DE = \sqrt{(8 - 7)^2 + (0 - 4)^2} = \sqrt{(1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17}\) The scale factor for the dilation is \(k = \dfrac{\sqrt{17}}{\sqrt{68}}= \sqrt{\dfrac{1}{4}}= \dfrac{1}{2}\). Using the scale factor, one possible location for \(F\) is at \((6, 0)\) because \(FE = 2\) and its corresponding length is \(BC = 4\). Checking the length from \(F\) to \(E\): \(FE= \sqrt{(6 - 7)^2 + (0 - 4)^2} = \sqrt{(-1)^2 + (-4)^2}\) \(= \sqrt{1 + 16} = \sqrt{17}\). |
Therefore \(\triangle ABC\sim\triangle DFE\) by \(SSS\sim\). Where is another possible location for \(F\)?
Quick Check
1) Determine if the following polygons are similar. If they are similar, write a similarity statement. If they are not similar, include an explanation why they are not similar. Please note: each polygon below is regular because all of its angles are congruent and all of its sides are congruent. |