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Chapter 14: Chapter 14

Problem 275

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\(\mathrm{D}\) and \(\mathrm{E}\) are respective points of side \(\underline{\mathrm{AB}}\) and \(\underline{\mathrm{BC}}\) of $\triangle \mathrm{ABC}\(, so that \)\mathrm{AD} / \mathrm{DB}=2 / 3\( and \)\mathrm{BE} / \mathrm{EC}=1 / 4 .\( If \)\underline{\mathrm{AE}}\( and \)\underline{D C}$ meet at \(\mathrm{P}\), find \(\mathrm{PC} / \mathrm{DP}\).

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The ratio PC/DP is 1:1, meaning the segments PC and DP have equal lengths.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Understand the problem and label the triangle

Draw triangle ABC with points D on AB and E on BC. Mark point P as the intersection of lines AE and DC. We are given the following ratios: AD/DB = 2/3 and BE/EC = 1/4. Our goal is to find the ratio PC/DP.

Step 2: Identify similar triangles

To solve this problem, we will need to identify similar triangles within the figure. Notice that triangles ADP and EDP are similar because they share angle D and angle ADE is congruent to angle EDE. Additionally, triangles BEP and ECP are similar because they share angle E and angle DEC is congruent to angle ECB.

Step 3: Determine ratios of corresponding sides for similar triangles

As triangles ADP and EDP are similar, we can write the following side ratios: \[\frac{AP}{EP} = \frac{AD}{ED}\] And for triangles BEP and ECP, \[\frac{BP}{EP} = \frac{BE}{EC}\]

Step 4: Use given ratios to express ratios in terms of AD and BE

We are given that AD/DB = 2/3 and BE/EC = 1/4. Therefore, \[ED = AD + DB = AD + \frac{3}{2}AD = \frac{5}{2}AD\] And, \[EC = BE + CD = BE + DB + DB = BE + 2DB = BE + (3DB - BE) = 3DB = 3(\frac{3}{2}AD) = \frac{9}{2}AD\] Substitute the values of ED and EC in the ratios found in step 3: \[\frac{AP}{EP} = \frac{AD}{\frac{5}{2}AD} = \frac{2}{5}\] \[\frac{BP}{EP} = \frac{BE}{\frac{9}{2}AD} = \frac{2}{9}\]

Step 5: Apply Ceva's theorem

Ceva's theorem states that for point P inside triangle ABC, lines AP, BP, and CP are concurrent (intersect at a single point) if and only if the ratio product: \[\frac{AP}{PB} \cdot \frac{BP}{PC} \cdot \frac{CP}{PA} = 1\] Using the values found in Step 4, we can determine the ratio PC/DP: \[\frac{AP}{EP} \cdot \frac{DP}{PC} \cdot \frac{CP}{AP} = 1\] \[\frac{2}{5} \cdot \frac{DP}{PC} \cdot \frac{CP}{AP} = 1\]

Step 6: Solve for PC/DP ratio

To find the value of PC/DP, we need to manipulate the equation from Step 5: \[\frac{DP}{PC} = \frac{5}{2} \cdot \frac{AP}{CP}\] Use the equation found in step 4: \[\frac{DP}{PC} = \frac{5}{2} \cdot \frac{2}{5}\] \[\frac{DP}{PC} = 1\] Therefore, the ratio PC/DP is 1:1, meaning the segments PC and DP have equal lengths.

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Most popular questions from this chapter

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