Both triangles are similar. Therefore, to find the scale factor, we calculate the ratio of corresponding side lengths.
DE/AB = EF/BC = FD/CA
⇓
5/20 = 4/16 = 3/12 = 1/4
From the above, we conclude that the scale factor is k= 14.
Question 2
Consider now the second question.
What is the ratio of their areas?
The Areas of Similar Polygons Theorem says that the ratio of the areas of two similar polygons is equal to the square of the scale factor.
Area of â–ł DEF/Area of â–ł ABC = k^2
Consequently, this question is asking for the square of the scale factor. From the previous question we know that k= 14.
Area of â–ł DEF/Area of â–ł ABC = ( 1/4)^2 = 1/16
Question 3
Let's consider the third question now.
What is the ratio of their corresponding side lengths?
Recall that the ratio of corresponding side lengths of two similar polygons is equal to the scale factor. Therefore, this question is asking for the scale factor between the given triangles, which is k= 14.
DE/AB = EF/BC = FD/CA
⇓
5/20 = 4/16 = 3/12 = 1/4
Question 4
Finally, consider the last given question.
What is the ratio of their perimeters?
The Perimeters of Similar Polygons Theorem says that the ratio of the perimeters of two similar polygons is equal to the scale factor of the similar polygons.
Perimeter ofâ–ł DEF/Perimeter ofâ–ł DEF = k
Therefore, this statement is asking for the scale factor. By the first question, we know that the scale factor is k= 14.
Perimeter ofâ–ł DEF/Perimeter ofâ–ł DEF = 1/4
Conclusion
In the following table, we summarize the results obtained before.
Question
Asks for
Answer
1
Scale factor
1/4
2
Scale factor squared
1/16
3
Scale factor
1/4
4
Scale factor
1/4
As we can see, the second question is different from questions 1, 3, and 4. The two required answers are 14 and 116.
