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Apply proportional reasoning to similar triangles

Similar triangles have equal corresponding angles and proportional corresponding side lengths, with a constant scale factor relating one triangle to the other even when their orientations differ. The learner interprets and sets up ratios between matching sides to find unknown lengths and solve simple geometric or indirect-measurement problems, avoiding comparisons of noncorresponding sides; the scope is numerical and straightforward algebraic rather than advanced transformation theory or trigonometric applications.

Detailed Explanation: Apply proportional reasoning to similar triangles

Similar triangles have the same shape, so their corresponding angles are equal and their corresponding sides are in the same ratio. The triangles may be turned or flipped, so match sides by their corresponding angles—not by their position on the page.

Suppose △ABC∼△DEF\triangle ABC \sim \triangle DEF, with the correspondence

A↔D,B↔E,C↔F.A \leftrightarrow D,\qquad B \leftrightarrow E,\qquad C \leftrightarrow F.

This means the matching sides are

AB↔DE,BC↔EF,AC↔DF.AB \leftrightarrow DE,\qquad BC \leftrightarrow EF,\qquad AC \leftrightarrow DF.

Example

In two similar triangles:

  • AB=6AB=6 cm
  • BC=9BC=9 cm
  • DE=10DE=10 cm

Find EFEF.

Step 1: Identify corresponding sides.

Since AB↔DEAB \leftrightarrow DE and BC↔EFBC \leftrightarrow EF, use those pairs. Do not compare ABAB with EFEF, because they are not corresponding sides.

Step 2: Find the scale factor.

The side ABAB grows from 66 cm to 1010 cm:

scale factor=DEAB=106=53.\text{scale factor}=\frac{DE}{AB}=\frac{10}{6}=\frac{5}{3}.

Step 3: Apply the same scale factor to the matching side.

The side BCBC corresponds to EFEF:

EF=BC(53)=9(53)=15.EF=BC\left(\frac{5}{3}\right) =9\left(\frac{5}{3}\right) =15.

Therefore,

EF=15 cm.\boxed{EF=15\text{ cm}}.

You can also set up a proportion directly:

DEAB=EFBC⇒106=EF9.\frac{DE}{AB}=\frac{EF}{BC} \quad\Rightarrow\quad \frac{10}{6}=\frac{EF}{9}.

Both methods give the same answer. Always make sure each ratio compares corresponding sides in the same order.

Learn by doing: Apply proportional reasoning to similar triangles

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