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Chapter 13: Chapter 13

Problem 246

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The lengths of the sides of a triangle are 6,8, and 12 . The lengths of the sides of a second triangle are \(1(1 / 2), 2\), and 3 . Are the two triangles similar?

Short Answer

Expert verified
The two triangles are similar because the ratios of their corresponding sides are equal: \(\frac{6}{1\frac{1}{2}} = \frac{8}{2} = \frac{12}{3} = 4\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Find the ratios of corresponding sides

Divide the length of each side of the first triangle by the corresponding side of the second triangle. Ratio 1: \(\frac{6}{1\frac{1}{2}} = \frac{6}{\frac{3}{2}}\) Ratio 2: \(\frac{8}{2}\) Ratio 3: \(\frac{12}{3}\)

Step 2: Simplify the ratios

Simplify the ratios found in Step 1. Ratio 1: To simplify this ratio, we will multiply the numerator and denominator by 2 to get rid of the fraction in the denominator. \[\frac{6}{\frac{3}{2}} = \frac{6 \times 2}{\frac{3}{2} \times 2} = \frac{12}{3}\] Ratio 2: This ratio can be simplified directly. \(\frac{8}{2} = 4\) Ratio 3: This ratio is already in simplest form. \(\frac{12}{3} = 4\)

Step 3: Compare the ratios

Now we will compare the simplified ratios. Ratio 1: \(\frac{12}{3} = 4\) Ratio 2: \(4\) Ratio 3: \(4\) All three ratios are equal to 4.

Step 4: Determine if the triangles are similar

Since all the corresponding side ratios are equal, the two triangles are indeed similar.

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Most popular questions from this chapter

Chapter 13

In the accompanying figure, points \(D, E\), and \(F\) are points on the triangle \(\triangle \mathrm{ABC}\) such that $\underline{\mathrm{AD}}, \underline{\mathrm{BE}}\(, and \)\underline{\mathrm{CF}}$ are concurrent at point P. Show that $$ (\mathrm{BD} / \mathrm{DC})(\mathrm{CE} / \mathrm{EA})(\mathrm{AF} / \mathrm{FB})=1 $$
Chapter 13

Show that a regular pentagon and the pentagon determined by joining all the vertices of the regular pentagon are similar.
Chapter 13

(a) In \(\triangle \mathrm{ABC}\), if \(\mathrm{D}\) is the midpoint of \(\underline{\mathrm{AB}}, \mathrm{E}\) is the midpoint of \(\underline{\mathrm{AC}}\), and \(\mathrm{F}\) is the midpoint of \(\underline{\mathrm{BC}}\), then prove that $\underline{\mathrm{DE}} \| \mathrm{BC}$, \(\underline{E P} \| \underline{A B}\), and \(D P \| A C\). (b) Prove that \(\triangle \mathrm{DEF} \sim \Delta \mathrm{CBA}\).
Chapter 13

Given; \(\underline{A D}\) is an angle bisector of \(\triangle \mathrm{ABC} ;\) point \(\mathrm{E}\) is on \(\underline{\mathrm{AD}}\) such that $\mathrm{AB} \cdot \mathrm{AC}=\mathrm{AD} \cdot \mathrm{AE}\(. Proves \)\phi \mathrm{B} \cong<\mathrm{AEC}$.
Chapter 13

Given the A.A.A. (Angle, Angle, Angle) Similarity Theorem, prove the A.A. (Angle, Angle) Similarity Theorem.
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