#1**+1 **

hi

you seem to be confused about similarity is that right? well it's as the word means things which are similar. A simple example of similar figures could be a scaled down model of a house compared to the actual house....all the features and dimensions of the actual house have been scaled down to make the model. in the case of triangles one triangle is an enlargement of the other eg all equilateral triangles are similar. For instance consider one equilateral triangle with all sides 2cm and a second one with all sides 6cmm. The second triangle can be thought of as an enlargement of the first with a scale factor of 3 because 2cm x 3 is 6cm. Notice however that all the angles in both triangles remain unchanged. These two properties then are the essence of similar triangles...firstly that the three angles of one triangle are equal to the corresponding angles in the other triangle and that the corresponding sides are in proportion ie each pair of corresponding sides are in the same ratio. So in a nutshell triangles are similar if they have the same shape but are of different sizes. Mathematicians have come up with some tests for similar triangles which you need to use, just like for proving congruent triangles, for reasons which we needn't go into. There are 3 main tests :

i. Three angles of one triangle are correspondingly equal to the three angles of another. Which in shorthand is known as the triangles are equiangular. Some people accept (AAA) but this notation is not generally accepted in formal exams.

ii. All three sides of one triangle are proportional to the three corresponding sides of another triangle. Thought of as (SSS).

iii. Two sides of one triangle are proportional to the corresponding pair of sides on another triangle and the included angles are equal. Thought of as (SAS).

The shortcuts (SSS) etc are commonly used except in formal exams.

A fourth test which is not widely accepted is for right triangles...(RHS)

Once you understand the basic definition and go through as many examples as you can things should become much clearer....one last thing...you may need to rotate or reflect the triangles in some cases in order to see that they are similar.

so for 13,14 and 15:

13) Basically shows the definition diagramatically which I cannot show you at the moment but someone else might.

14) a and d are true

15) In triangle ABC, triangle DEF AC/DE = 10/15 AB/DF = 6/9 and BC/EF = 8/12

= 2/3 = 2/3 = 2/3

Therefore triangle ABC is similar to triangle DEF

For 11) and 12) with quadrilaterals also you can check that corresponding pairs of sides are in the same ratio and that they are equiangular. Here the rectangles all are 90 degrees but that is not enough....check that corresponding pairs of sides are also in proportion. Remember: same shape but different size and not all rectangles are the same shape regarding similarity whereas all square are similar as they all have right angles AND pairs of corresponding sides are in proportion.

hope this helps and hasn't confused you even more!!!??

Guest Feb 19, 2018

#2**+1 **

Before starting any problems listed above, you should understand what a similar polygon means. In general, it means that all corresponding angles in two figures are congruent and that all corresponding sides are proportional. Proving corresponding sides proportional requires that you find the ratio of corresponding sides and check for equality.

11)

In this example, both figures are rectangles, so every angle is a right angle. All right angles are congruent. The next step requires us to compare the ratio of the sides.

\(\frac{20}{40}\stackrel{?}=\frac{24}{48}\)

Notice what has happened here. For the smaller rectangle, the leftward fraction represents the ratio of the shorter side of the smaller rectangle to the longer side. The ratio is the same for the larger rectangle. Both fractions can be simplified.

\(2=2\)

The ratios are the same, so both rectangles are similar polygons.

12)

For this problem, simply employ the same strategy as before. The figures are rectangles, so it is already established that the angles are congruent. Now, let's compare the ratios of corresponding sides.

\(\frac{8}{16}\stackrel{?}= \frac{12}{30}\\ \frac{1}{2}\neq \frac{2}{5}\)

After simplifying both fractions to simplest terms, it becomes clear that the ratio of corresponding side lengths is not proportional. By definition, corresponding side lengths must be proportional in order for two figures to be considered similar. Since this is not the case, these rectangles are not similar.

13)

The diagram is not the prettiest work because I cannot figure out how to show congruent angles on this program. I used slipshod cut-off semicircles.

In the diagram above, \(\angle A\cong \angle X\text{ and }\angle B\cong \angle Y\text{ and }\angle C\cong\angle Z\). Also, \(\frac{2}{4}=\frac{3}{6}=\frac{4}{8}\), so \(\triangle ABC\sim\triangle XYZ\).

14) The beauty of a similarity (or congruence) statement is that it reveals more information than it may first appear. We should not forget what similarity means! Let's look at the individual choices.

\(m\angle C=m\angle Z\) is true because both letters appear in the same position in the similarity statement \(\triangle AB\textcolor{red}{C}\sim\triangle XY\textcolor{red}{Z}\). Corresponding angles are always congruent in similar triangles.

\(\frac{\textcolor{red}{AB}}{\textcolor{red}{XY}}=\frac{\textcolor{blue}{YZ}}{\textcolor{blue}{BC}}\) is not true! Notice that \(\triangle \textcolor{red}{AB}C \sim \triangle \textcolor{red}{XY}Z\text{ and }\triangle A\textcolor{blue}{BC}\sim\triangle X\textcolor{blue}{YZ}\). Remember that both ratios must compare a pair of corresponding sides. In the leftward statement, it compares a side of the smaller triangle to a corresponding side of the larger triangle. In the rightward statement, the ratio compares the length of the larger triangle to the shorter one. Notice how the similarity statement shows this discrepancy; notice that the letters are not in the same order.

\(AB\cong XY\) is not true. Yes, it lines up with the similarity statement \(\triangle \textcolor{red}{AB}C \sim \triangle \textcolor{red}{XY}Z\), but corresponding sides are **proportional**--not congruent.

\(\frac{\textcolor{red}{BC}}{\textcolor{red}{YZ}}=\frac{\textcolor{blue}{AC}}{\textcolor{blue}{XZ}}\) is true because it matches perfectly with the similarity statement. Notice that \(\triangle A\textcolor{red}{BC}\sim\triangle X\textcolor{red}{YZ}\) and \(\triangle \textcolor{blue}{A}B\textcolor{blue}{C}\sim\triangle \textcolor{blue}{X}Y\textcolor{blue}{Z}\).

15)

6-8-10 and 9-12-15 are a special set of numbers known as Pythagorean triples, which are positive-integer side lengths of a right triangle. Since both side lengths have this property, we can conclude that \(m\angle B=m\angle F= 90^\circ\). An accepted method of proving triangles similar is called Side-Angle-Side Similarity Theorem, which states that the one pair of sides is proportional, and the included angle is congruent.

\(\frac{6}{8}\stackrel{?}=\frac{9}{12}\\ \frac{3}{4}=\frac{3}{4}\)

The side lengths are proportional, so we can conclude, by the Side-Angle-Side Triangle Similarity Theorem, that \(\triangle ABC\sim\triangle DFE\).

TheXSquaredFactor
Feb 19, 2018