3.6: Similar Triangles
- Page ID
- 174258
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definitions and Theorems
Two triangles are said to be congruent if their corresponding angles are equal, and their corresponding sides have the same lengths. Symbolically, we use the congruency symbol \( \cong \) to state that two triangles are congruent.
Two triangles that have the same shape—but not necessarily the same size—are similar. Similarity is the geometric basis for proportional reasoning about figures.
Two triangles are similar if their corresponding angles are congruent and the lengths of their corresponding sides are proportional. If \(\triangle ABC\) is similar to \(\triangle DEF\), you write \(\triangle ABC \sim \triangle DEF\), where the order of the vertices specifies the correspondence: \(A\) with \(D\), \(B\) with \(E\), and \(C\) with \(F\).
The symbol \(\sim\) is read as "is similar to." Thus \(\triangle ABC \sim \triangle DEF\) is read as "triangle \(ABC\) is similar to triangle \(DEF\)."
For two similar triangles, the scale factor is the constant ratio between the lengths of corresponding sides. If \(\triangle ABC \sim \triangle DEF\), the scale factor of \(\triangle ABC\) to \(\triangle DEF\) is the common value\[k = \dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{CA}{FD}.\nonumber\]
If \(\triangle ABC \sim \triangle DEF\), then the corresponding angles are congruent,\[\angle A \cong \angle D, \quad \angle B \cong \angle E, \quad \angle C \cong \angle F,\nonumber\]and the corresponding sides are proportional,\[\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{CA}{FD}.\nonumber\]
Two triangles are similar whenever any one of the following conditions holds:
- (AA) Two angles of one triangle are congruent to two angles of the other. The third pair of angles is then automatically congruent.
- (SSS) All three pairs of corresponding sides are proportional.
- (SAS) Two pairs of corresponding sides are proportional, and the angles included between those sides are congruent.
Examples
Suppose \(\triangle ABC \sim \triangle DEF\) with \(AB = 6\), \(BC = 9\), \(CA = 12\), and \(DE = 4\). Find the scale factor of \(\triangle ABC\) to \(\triangle DEF\), and find the lengths \(EF\) and \(FD\).
- Solution
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By the correspondence, \(AB\) pairs with \(DE\), so the scale factor is\[k = \dfrac{AB}{DE} = \dfrac{6}{4} = \dfrac{3}{2}.\nonumber\]Because every pair of corresponding sides shares this ratio, set each remaining pair equal to \(\tfrac{3}{2}\). For \(EF\):\[\dfrac{BC}{EF} = \dfrac{3}{2} \quad \Longrightarrow \quad \dfrac{9}{EF} = \dfrac{3}{2} \quad \Longrightarrow \quad EF = \dfrac{9 \cdot 2}{3} = 6.\nonumber\]For \(FD\):\[\dfrac{CA}{FD} = \dfrac{3}{2} \quad \Longrightarrow \quad \dfrac{12}{FD} = \dfrac{3}{2} \quad \Longrightarrow \quad FD = \dfrac{12 \cdot 2}{3} = 8.\nonumber\]Thus \(EF = 6\) and \(FD = 8\). As a check, \(\triangle DEF\) has sides \(4, 6, 8\), each exactly \(\tfrac{2}{3}\) of the corresponding side of \(\triangle ABC\).
Suppose \(\triangle ABC \sim \triangle DEF\) with \(AB = 8\), \(DE = 12\), and \(BC = 5\). Find the length \(EF\).
- Solution
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Let \(x = EF\). Pairing corresponding sides according to the similarity statement gives\[\dfrac{AB}{DE} = \dfrac{BC}{EF} \quad \Longrightarrow \quad \dfrac{8}{12} = \dfrac{5}{x}.\nonumber\]Cross-multiplying yields\[8x = 12 \cdot 5 = 60 \quad \Longrightarrow \quad x = \dfrac{60}{8} = 7.5.\nonumber\]Therefore \(EF = 7.5\).
One triangle has side lengths \(6\), \(8\), and \(10\). A second triangle has side lengths \(9\), \(12\), and \(15\). Are the triangles similar?
- Solution
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Pair the sides from smallest to smallest, and compare the three ratios:\[\dfrac{6}{9} = \dfrac{2}{3}, \quad \dfrac{8}{12} = \dfrac{2}{3}, \quad \dfrac{10}{15} = \dfrac{2}{3}.\nonumber\]All three pairs of corresponding sides are proportional with the same ratio \(\tfrac{2}{3}\), so the triangles are similar by the SSS criterion.
In \(\triangle ABC\), point \(D\) lies on side \(AB\) and point \(E\) lies on side \(AC\), with \(\overline{DE}\) parallel to \(\overline{BC}\). Given \(AD = 4\), \(DB = 6\), and \(AE = 5\), find the length \(EC\).
- Solution
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Because \(\overline{DE}\) is parallel to \(\overline{BC}\), the parallel lines create equal corresponding angles, so \(\angle ADE \cong \angle ABC\) and \(\angle AED \cong \angle ACB\). The two triangles also share \(\angle A\). By the AA criterion, \(\triangle ADE \sim \triangle ABC\).
Let \(x = EC\). The full sides are \(AB = AD + DB = 4 + 6 = 10\) and \(AC = AE + EC = 5 + x\). Setting corresponding sides proportional gives\[\dfrac{AD}{AB} = \dfrac{AE}{AC} \quad \Longrightarrow \quad \dfrac{4}{10} = \dfrac{5}{5 + x}.\nonumber\]Cross-multiplying,\[4(5 + x) = 50 \quad \Longrightarrow \quad 20 + 4x = 50 \quad \Longrightarrow \quad x = \dfrac{30}{4} = 7.5.\nonumber\]Therefore \(EC = 7.5\).
At the same time of day, a vertical flagpole casts a shadow \(24\) feet long, while a vertical \(6\)-foot post casts a shadow \(4\) feet long. Find the height of the flagpole.
- Solution
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Each object and its shadow form a right triangle. The sun's rays strike both objects at the same angle, so those angles are congruent, and each object meets the ground at a right angle. With two pairs of congruent angles, the triangles are similar by the AA criterion.
Let \(x\) be the height of the flagpole. Pairing each height with its own shadow length,\[\dfrac{x}{24} = \dfrac{6}{4}.\nonumber\]Solving,\[x = 24 \cdot \dfrac{6}{4} = 24 \cdot \dfrac{3}{2} = 36.\nonumber\]The flagpole is \(36\) feet tall.
Sources
Several parts of this text use modifications from the following source:
- Wikipedia article: "Similarity (geometry)"
This source is released under the Creative Commons Attribution-Share-Alike License 4.0.