SSS and SAS Similarity

Show your work for each problem.

1.
Which triangle is similar to a triangle with sides $2$ , $3$ and $4$ ?
1. a)4; 6; 8
2. b)4; 5; 6
3. c)3; 4; 5
4. d)4; 6; 9
Answer: 4; 6; 8
$\frac{4}{2} = \frac{6}{3} = \frac{8}{4} = 2$ , so the triangle with sides $4$ , $6$ , $8$ is similar by SSS Similarity with $k = 2$ . For $3$ , $4$ , $5$ the ratios are $1.5$ , $1.33$ , $1.25$ , for $4$ , $5$ , $6$ they are $2$ , $1.67$ , $1.5$ , and for $4$ , $6$ , $9$ they are $2$ , $2$ , $2.25$ .
2.
$\triangle ABC$ has sides $5$ , $7$ , $8$ and $\triangle DEF$ has sides $15$ , $21$ , $24$ . The triangles are similar. What is the scale factor $k$ from $\triangle ABC$ to $\triangle DEF$ ?
Answer: 3
$\frac{15}{5} = \frac{21}{7} = \frac{24}{8} = 3$ , so $k = 3$ .
3.
A triangle has sides $3$ , $5$ , $6$ and another has sides $9$ , $15$ , $18$ . Check all three ratios with SSS Similarity and find the scale factor.
Answer: 3
1. What is the ratio of the shortest sides, $\frac{9}{3}$ ? 3
2. What is the ratio of the middle sides, $\frac{15}{5}$ ? 3
3. What is the ratio of the longest sides, $\frac{18}{6}$ ? 3
4. Are all three ratios equal, so that the triangles are similar? Type 1 for yes or 0 for no. 1
4.
For $\triangle ABC$ and $\triangle DEF$ you know $\frac{DE}{AB} = \frac{DF}{AC} = 2$ and $\angle A = \angle D$ . Which criterion proves that the triangles are similar?
1. a)SAS similarity
2. b)AA similarity
3. c)SSS similarity
4. d)Not enough information
Answer: SAS similarity
Two pairs of sides have the same ratio $2$ , and $\angle A$ is the included angle of $AB$ and $AC$ ( $\angle D$ of $DE$ and $DF$ ). That is SAS Similarity. AA needs two angles, SSS needs three sides.
5.
$\triangle ABC \sim \triangle DEF$ with scale factor $k = 2$ from $\triangle ABC$ to $\triangle DEF$ . If $AB = 6$ , how long is $DE$ ?
Answer: 12
$DE = k \cdot AB = 2 \cdot 6 = 12$ .
6.
$\triangle ABC$ has sides $8$ , $4$ , $6$ and $\triangle DEF$ has sides $9$ , $12$ , $6$ . Pair the sides correctly and find the scale factor from $\triangle ABC$ to $\triangle DEF$ .
Answer: 1.5
1. What is the shortest side of $\triangle ABC$ ? 4
2. What is the shortest side of $\triangle DEF$ ? 6
3. What is the ratio of the shortest sides, $\frac{6}{4}$ ? 1.5
4. Check with the longest sides: what is $\frac{12}{8}$ ? 1.5
7.
Two triangles are similar with scale factor $k = 3$ . The smaller one has an area of $5$ cm². What is the area of the larger one?
1. a)45 cm²
2. b)30 cm²
3. c)15 cm²
4. d)9 cm²
Answer: 45 cm²
The area ratio is $k^2 = 3^2 = 9$ , so the area is $5 \cdot 9 = 45$ cm². $15$ cm² would be the mistake of using $k$ instead of $k^2$ .
8.
Two similar triangles have scale factor $k = 2.5$ from the smaller to the larger one. The smaller triangle has a perimeter of $18$ cm. What is the perimeter of the larger triangle in cm?
Answer: 45
Perimeters grow by the same factor as the sides: $18 \cdot 2.5 = 45$ cm.
9.
$\triangle ABC$ : $AB = 3$ , $AC = 5$ , $\angle A = 120°$ , $BC = 7$ . $\triangle DEF$ : $DE = 9$ , $DF = 15$ , $\angle D = 120°$ . Show that the triangles are similar and find $EF$ .
Answer: 21
1. What is $\frac{DE}{AB}$ ? 3
2. What is $\frac{DF}{AC}$ ? 3
3. Is $\angle A$ the included angle of $AB$ and $AC$ ? Type 1 for yes or 0 for no. 1
4. The triangles are similar by SAS with $k = 3$ . What is $EF = 3 \cdot BC$ ? 21
10.
$\triangle ABC$ : $AB = 6$ , $AC = 9$ , $\angle A = 40°$ . $\triangle DEF$ : $DE = 8$ , $DF = 12$ , $\angle D = 40°$ . What can you conclude?
1. a)Not enough information
2. b)Similar by SSS Similarity, $k = \frac{4}{3}$
3. c)Similar by SAS Similarity, $k = \frac{4}{3}$
4. d)Not similar, the side ratios are different
Answer: Similar by SAS Similarity, $k = \frac{4}{3}$
$\frac{8}{6} = \frac{4}{3}$ and $\frac{12}{9} = \frac{4}{3}$ , and the $40°$ angles lie between these sides. So the triangles are similar by SAS Similarity with $k = \frac{4}{3}$ . SSS is not possible because the third sides are not given.
11.
$\triangle ABC$ has sides $6$ , $8$ , $10$ . A similar triangle has sides $9$ , $12$ and $x$ , where $x$ is its longest side. Find $x$ .
Answer: 15
$k = \frac{9}{6} = \frac{12}{8} = 1.5$ , so $x = 10 \cdot 1.5 = 15$ .
12.
The triangles with sides $4$ , $6$ , $9$ and $6$ , $x$ , $13.5$ are similar (sides listed from shortest to longest). Find $x$ .
Answer: 9
1. What is $k$ from the shortest sides, $\frac{6}{4}$ ? 1.5
2. Check with the longest sides: what is $\frac{13.5}{9}$ ? 1.5
3. What is the middle side $x = 6 \cdot k$ ? 9
13.
In $\triangle ABC$ , point $D$ lies on $AB$ and point $E$ on $AC$ . $AD = 3$ , $AB = 9$ , $AE = 4$ , $AC = 12$ and $DE = 5$ . Find $BC$ .
Answer: 15
1. What is $\frac{AD}{AB}$ ? Write it as a fraction. 1/3
2. What is $\frac{AE}{AC}$ ? Write it as a fraction. 1/3
3. By SAS, $\triangle ADE \sim \triangle ABC$ . What is the scale factor from $\triangle ADE$ to $\triangle ABC$ ? 3
4. What is $BC = 3 \cdot DE$ ? 15
14.
$\triangle ABC$ : $AB = 4$ , $BC = 6$ , $\angle C = 30°$ . $\triangle DEF$ : $DE = 8$ , $EF = 12$ , $\angle F = 30°$ . Are the triangles similar?
1. a)Yes, by SAS Similarity
2. b)No, the side ratios are different
3. c)Cannot be decided: the angle is not between the two given sides
4. d)Yes, by SSS Similarity
Answer: Cannot be decided: the angle is not between the two given sides
The side ratios are both $2$ , but the angle between $AB$ and $BC$ is $\angle B$ , not $\angle C$ . Two sides and a non-included angle (SSA) do not fix the shape: here $\angle A$ could be about $49°$ or about $131°$ . So SAS Similarity does not apply and you cannot decide.
15.
$\triangle ABC$ : $AB = 8$ , $BC = 12$ , $\angle B = 55°$ . $\triangle DEF$ : $DE = 6$ , $EF = x$ , $\angle E = 55°$ . For which value of $x$ are the triangles similar by SAS Similarity (with $AB \leftrightarrow DE$ and $BC \leftrightarrow EF$ )?
Answer: 9
$k = \frac{DE}{AB} = \frac{6}{8} = 0.75$ . The included angles $B$ and $E$ are equal, so we need $\frac{x}{12} = 0.75$ , which gives $x = 9$ .
16.
$\triangle ABC$ has sides $5$ , $12$ , $13$ and a right angle between the sides $5$ and $12$ . A similar triangle $\triangle DEF$ has shortest side $15$ . Find the perimeter and then the area of $\triangle DEF$ .
Answer: 270
1. What is the scale factor $k$ from $\triangle ABC$ to $\triangle DEF$ ? 3
2. What is the perimeter of $\triangle ABC$ ? 30
3. What is the perimeter of $\triangle DEF$ ? 90
4. What is the area of $\triangle ABC$ ? (The legs $5$ and $12$ are base and height.) 30
5. What is the area of $\triangle DEF$ ? 270
17.
In $\triangle ABC$ , point $D$ lies on $AB$ and point $E$ on $AC$ . $AD = 6$ , $DB = 4$ , $AE = 9$ , $EC = 6$ and $DE = 12$ . Show that $\triangle ADE \sim \triangle ABC$ and find $BC$ .
Answer: 20
1. How long is the whole side $AB$ ? 10
2. How long is the whole side $AC$ ? 15
3. What is $\frac{AD}{AB}$ as a decimal? 0.6
4. What is $\frac{AE}{AC}$ as a decimal? 0.6
5. Same ratio and shared angle $A$ : SAS. What is $BC = \frac{DE}{0.6}$ ? 20
18.
Two similar triangles have areas $20$ cm² and $45$ cm². The shortest side of the smaller triangle is $4$ cm. How long is the shortest side of the larger triangle, in cm?
Answer: 6
1. What is the area ratio $\frac{45}{20}$ ? 2.25
2. The area ratio is $k^2$ . What is $k$ ? 1.5
3. What is the shortest side of the larger triangle, $4 \cdot k$ ? 6