Similar Polygons

Similar polygons room two polygons v the very same shape, yet not the very same size. Similar polygons have corresponding angles that room congruent, and corresponding political parties that space proportional.

You are watching: Are the polygons similar if they are write a similarity statement and give the scale factor

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Figure (PageIndex1)

These polygons room not similar:

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Solution

All the matching angles space congruent due to the fact that the forms are rectangles.

Let’s see if the sides room proportional. (dfrac812=dfrac23) and also (dfrac1824=dfrac34). (dfrac23 eq dfrac34), therefore the sides room not in the same proportion, and the rectangles space not similar.


Example (PageIndex2)

(Delta ABCsim Delta MNP). The perimeter that (Delta ABC) is 150, (AB=32) and (MN=48). Find the perimeter the (Delta MNP).

Solution

From the similarity statement, (AB) and (MN) are corresponding sides. The scale aspect is (dfrac3248=dfrac23) or (dfrac32). Delta ABC) is the smaller triangle, so the perimeter that (Delta MNP) is (dfrac32(150)=225).


Example (PageIndex3)

Suppose (Delta ABCsim Delta JKL). Based upon the similarity statement, i beg your pardon angles room congruent and also which sides space proportional?

Solution

Just favor in a congruence statement, the congruent angles line up within the similarity statement. So, (angle Acong angle J), (angle Bcong angle K), and also angle Ccong angle L). Write the political parties in a proportion: (dfracABJK=dfracBCKL=dfracACJL). Note that the proportion might be created in different ways. Because that example, (dfracABBC=dfracJKKL) is likewise true.


Example (PageIndex4)

(MNPQ sim RSTU). What space the values of (x), (y) and also (z)?

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Figure (PageIndex4)

Solution

In the similarity statement, (angle Mcong angle R), so (z=115^circ). For (x) and also (y), collection up proportions.

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(dfrac1830=dfracx25 qquad dfrac1830=dfrac15y)

(450=30x qquad 18y=450)

(x=15qquad y=25)


Example (PageIndex5)

(ABCDsim AMNP). Find the range factor and also the size of (BC).

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