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Identify similar figures

Similar figures have the same shape: corresponding angles are congruent, and corresponding side lengths are proportional by one constant scale factor, which may enlarge or reduce the figure; congruent figures are the special case with scale factor 1. In coordinate representations, this relationship can be identified through a dilation, possibly combined with a translation, rotation, or reflection, and by matching corresponding vertices and lengths. Formal similarity proofs and more advanced transformations are not included.

Detailed Explanation: Identify similar figures

Figures are similar if they have the same shape. To check, look for:

  1. Matching corresponding vertices and angles.
  2. A single scale factor that changes every side length by the same amount.
  3. In a coordinate plane, a dilation may be combined with a translation, rotation, or reflection.

A scale factor greater than 11 enlarges a figure. A scale factor between 00 and 11 reduces it. A scale factor of 11 means the figures are congruent.

Example

Are â–³ABC\triangle ABC and â–³DEF\triangle DEF similar?

A(1,1),B(3,1),C(1,2)A(1,1),\quad B(3,1),\quad C(1,2) D(4,2),E(8,2),F(4,4)D(4,2),\quad E(8,2),\quad F(4,4)

Step 1: Match the corresponding vertices

The horizontal and vertical positions show this matching:

  • AA corresponds to DD
  • BB corresponds to EE
  • CC corresponds to FF

Step 2: Compare corresponding side lengths

For â–³ABC\triangle ABC:

  • AB=2AB=2
  • AC=1AC=1
  • BC=(3−1)2+(1−2)2=5BC=\sqrt{(3-1)^2+(1-2)^2}=\sqrt{5}

For â–³DEF\triangle DEF:

  • DE=4DE=4
  • DF=2DF=2
  • EF=(8−4)2+(2−4)2=20=25EF=\sqrt{(8-4)^2+(2-4)^2}=\sqrt{20}=2\sqrt{5}

Now compare each pair:

DEAB=42=2\frac{DE}{AB}=\frac{4}{2}=2 DFAC=21=2\frac{DF}{AC}=\frac{2}{1}=2 EFBC=255=2\frac{EF}{BC}=\frac{2\sqrt{5}}{\sqrt{5}}=2

All corresponding sides have the same scale factor, 22.

Step 3: Check the coordinate relationship

Multiplying the coordinates of â–³ABC\triangle ABC by 22 gives:

(1,1)→(2,2),(3,1)→(6,2),(1,2)→(2,4)(1,1)\to(2,2),\quad (3,1)\to(6,2),\quad (1,2)\to(2,4)

Moving each point 22 units to the right gives:

(2,2)→(4,2),(6,2)→(8,2),(2,4)→(4,4)(2,2)\to(4,2),\quad (6,2)\to(8,2),\quad (2,4)\to(4,4)

These are exactly DD, EE, and FF.

Therefore, â–³ABC\triangle ABC and â–³DEF\triangle DEF are similar. The second triangle is an enlargement of the first by a scale factor of 22, followed by a translation.

Learn by doing: Identify similar figures

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    Similar Triangles - Separate (No Rotation) to Side

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