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Determine if the following polygons are similar.

Determine If The Following Polygons Are Similar class=
Determine If The Following Polygons Are Similar class=

Sagot :

Problem 1

Triangle ABE is similar to triangle DCE. This is because we have two pairs of congruent corresponding angles.

  • angle AEB = angle DEC (vertical angles)
  • angle ABE = angle DCE (alternate interior angles)

Refer to the angle angle (AA) similarity theorem for more info.

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For the other pair of triangles, we don't have enough info to determine if they are similar or not. We have info about one pair of sides, and a pair of angles, but that's about it.

We need to know if JL is proportional to MO, and in the same ratio as KL & NO are, if we wanted to use the side angle side (SAS) similarity theorem.

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Summary:

  • Triangle ABE is similar to triangle DCE by the AA (angle angle) similarity theorem
  • We don't have enough info to know if triangles JKL and MNO are similar or not.

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Problem 2

Divide the corresponding sides

  • LN/PO = 30/15 = 2
  • LN/OQ= 26/13 = 2
  • ML/QP = 26/13 = 2

We see the three ratios are equal to the same value (2) and that proves the triangles LNM and POQ are similar by the SSS (side side side) similarity theorem.

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Like before, we'll divide the corresponding sides

  • 1st  Rectangle: horizontal/vertical = 10/4 = 2.5
  • 2ndRectangle: horizontal/vertical = 3/8 = 0.375

We don't have a match. Let's try rotating the second rectangle so that the side '8' is horizontal, and the side '3' is vertical

  • 1st Rectangle: horizontal/vertical = 10/4 = 2.5
  • 2nd Rectangle: horizontal/vertical = 8/3 = 2.67 approximately

We don't have a match here either. There are other approaches we could take, but basically the two rectangles are not similar.

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Summary:

  • Triangles LNM and POQ are similar by the SSS (side side side) similarity theorem.
  • The rectangles are not similar.
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