Similar Triangles
When two triangles have the same ratio of corresponding sides or congruent corresponding angles, they are said to be similar. The \(‘\sim’ \) symbol here denotes the similarity of triangles.
In the above figure, two triangles \(\triangle~PQR \) and \(\triangle~XYZ \) are similar if,
i) \(\angle~P~=~ \angle~X,~\angle~Q~=~\angle~Y\) and \(\angle~R~=~ \angle~Z \). Always remember if two pairs of angles are congruent, then the third pair is also congruent.
ii) \(\frac{PQ}{XY}~=~ \frac{QR}{YZ}~=~\frac{PR}{XZ}\) (Similar triangles have proportional sides)
As a result, if the above-mentioned conditions are met, we can conclude that \(\triangle~PQR~\sim ~\triangle~XYZ\)
For example, Identifying the similar triangles
As we know, the corresponding sides of both triangles should be proportional to one another.
That is, \(\frac{PR}{PQ}~=~\frac{XZ}{XY}\)
\(\frac{PR}{PQ}~=~\frac{10}{4}~=~\frac{2\times~5}{2\times~2}~=~\frac{5}{2}\)
Check if, \(\frac{PR}{PQ}~=~\frac{XZ}{XY}\)
\(\frac{10}{4}~=~\frac{5}{2}\)
Therefore the triangles are similar as the ratios of their sides are equal.
For example, Identifying the similar triangles.
Writing and solving the equation to find the value of \(x\) and \(y\).
As we know, the sum of the angles within a triangle is equal to 180 degrees.
\(x+ 39+34~=~180\)
\(x+ 73~=~180\)
\(x~= 180~-~73\)
\(x~= 107\)
Writing and solving the equation to find the value of \(y\) .
As we know, the sum of the angles within a triangle is equal to 180 degrees.
\(x+ 39+107~=~180\)
\(x+ 146~=~180\)
\(x~= 180~-146\)
\(x =~34\)
Because the corresponding angles of both triangles are not equal, the two triangles are not similar.
Finding missing measures of Similar Triangles
What if you were standing close to a tower and wondered about its height? How could you determine the height of the building using your own height, the length of the shadow projected by you and the tower?
Similar triangles can be used to indirectly measure lengths. This method can be used to determine the width of a river as well as the height of a tall object.
For example, Finding the missing measure of the similar triangles given below.
Corresponding sides should be proportional to one another.
Therefore, \(\frac{PR}{PQ}~=~\frac{XZ}{XY}\)
Let XY be x
\(\frac{16}{10}~=~\frac{8}{x}\)
\(16x~=~8\times~10\) (cross multiplication)
\(x~=~\frac{8\times~10}{16}\)
\(x~=~5\) (simplified)
Therefore the missing measurement \(x~=~5\).
Examples
1. Identifying similar triangles.
Solution:
As we know, Both triangles, corresponding sides should be proportional to one another for the triangles to be similar.
Therefore,
\(\frac{2}{4}~=~\frac{3}{6}~=~\frac{4}{8}~=~\frac{1}{2}\) (simplified)
Therefore the triangles are similar as the ratios are equal.
2. David and Joseph cut the two papers into two triangles. Find whether the triangles are similar or not.
Solution: As we know, the corresponding sides of both triangles should be proportional to one another for the triangles to be similar.
Let’s find the ratio of each pair of corresponding sides
\(\frac{10}{6}~=~\frac{10\div~2}{6\div~2}~=~\frac{5}{3}~=~\frac{1}{2}\)
\(\frac{12}{5}\)
\(\frac{16}{8}~=~\frac{16\div~8}{8\div~8}~=~\frac{2}{1}\)
\(\frac{5}{3}~\neq ~\frac{12}{5}~\neq ~\frac{2}{1}\)
Therefore the triangles are not similar as the ratios of sides are not equal.
3. Data of the given figure is \(CE~=~50,~EF~=~35\), and \(CD~=~(3x~+~12),~DG~=~21\). Find the value of \(‘x’\).
Solution:
In \(\triangle~GCD\) and \(\triangle~FCE\)
\(\angle~D~=~\angle~E\) (Given that both angles are 90 degrees)
\(\angle~C~=~\angle~C\) (\(\angle~C\) is common to both triangles)
Since two pairs of angles are equal, the third pair is also equal. Since all corresponding angles are congruent to each other;
\(\triangle~GCD~\sim ~\angle~FCE\)
The corresponding sides of both triangles should be proportional to one another.
Therefore, \(\frac{CD}{DG}~=~\frac{CE}{EF}\)
\(\frac{3x+12}{21}~=~\frac{50}{35}\)
\(35~(3x+12)~=~50\times~21\) (cross multiplication)
\(105x+420~=~1050\) (using properties of multiplication)
\(105x~=~630\) (using division)
\(x~=~\frac{630}{105}\)
\(x~=~6\)
Therefore, the value of \(x\) is 6.
4. A four-foot-tall person casts a two-foot shadow on a sunny day, this is proportional to the 8-foot shadow cast by a nearby flagpole. What is the height of the flagpole?
Solution: Because it is given that the lengths are proportional,
Therefore, \(\frac{8}{x}~=~\frac{2}{4}\)
\(32~=~2\times~x\) (cross multiplication)
\(x~=~\frac{32}{2}\)
\(x~=~16\)
Therefore the height of the flagpole is 16 feet.
When two triangles have equal angles or proportional sides they are said to be similar.
When two triangles are similar, the ratio of their areas is the ratio of the square of their corresponding sides.
Yes, all identical figures will have the same angles and their sides will have a ratio of 1, as all sides are equal. Hence, they will be similar but all similar figures need not be identical.