Similar Polygons

Similar polygons room two polygons with the same shape, but not the same size. Comparable polygons have corresponding angles that room congruent, and corresponding sides that are proportional.

You are watching: The polygons are similar but not

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Figure (PageIndex1)

These polygons room not similar:

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Solution

All the matching angles room congruent because the shapes are rectangles.

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


Example (PageIndex2)

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

Solution

From the similarity statement, (AB) and also (MN) are equivalent sides. The scale element is (dfrac3248=dfrac23) or (dfrac32). Delta ABC) is the smaller sized triangle, so the perimeter of (Delta MNP) is (dfrac32(150)=225).


Example (PageIndex3)

Suppose (Delta ABCsim Delta JKL). Based on the similarity statement, which angles are congruent and which sides room proportional?

Solution

Just like in a congruence statement, the congruent angle line up within the similarity statement. So, (angle Acong angle J), (angle Bcong angle K), and also angle Ccong angle L). Write the sides in a proportion: (dfracABJK=dfracBCKL=dfracACJL). Note that the proportion might be written in different ways. For example, (dfracABBC=dfracJKKL) is additionally true.


Example (PageIndex4)

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

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Figure (PageIndex4)

Solution

In the similarity statement, (angle Mcong angle R), therefore (z=115^circ). Because that (x) and (y), collection up proportions.

See more: 2004 Pontiac Grand Prix Power Steering Problems, Power Steering Issues

(dfrac1830=dfracx25 qquad dfrac1830=dfrac15y)

(450=30x qquad 18y=450)

(x=15qquad y=25)


Example (PageIndex5)

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

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