Maintaining Mathematical Proficiency - 4. Transformations

If figures are similar, they must have two things in common.

The figures must have congruent angles. This means that the angles of the first figure should be the same as the angles of the second figure.
The figures must have the same proportions of their side lengths.

Are the Angles Congruent?

Let's consider the given figures.

Both of the given figures are rectangles. Therefore, they both have four right angles. Because all of the angles are the same, we know that they are congruent.

Are the Side Lengths Proportional?

Now we need to check if the side lengths are proportional. We need the ratio of the lengths to be equal to the ratio of the widths.

Let's consider the lengths of the rectangles to be the longer sides and the widths to be the shorter sides. Note that we've named the rectangles A and B for easier reference. Also, it does not matter which rectangle we put in the numerator of the ratio as long as we do the same thing for both measurements. Length of BLength of A= Width of BWidth of A Let's begin by calculating the ratio between the lengths.

Length of B/Length of A

SubstituteII

Length of B= 14, Length of A= 12

14/12

ReduceFrac

a/b=.a /2./.b /2.

7/6

The ratio between the lengths is 76. Let's do the same thing for the widths.

Width of B/Width of A

SubstituteII

Width of B= 7, Width of A= 5

7/5

The ratio between the widths is 75. The ratios are not the same. 76 ≠ 75

Conclusion

Because the ratios between the measurements of the rectangles are not the same, they cannot be similar.

Exercise 5 Page 171

Hints

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Practice exercises

Asses your skill

Are the Angles Congruent?

Are the Side Lengths Proportional?

Conclusion