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Problem 275

\(\mathrm{D}\) 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 \(\mathrm{PC} / \mathrm{DP}\).

Expert verified

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

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Chapter 14

Given: \(\mathrm{CD}\) is an altitude and \(\mathrm{CE}\) is an angle bisector of \(\triangle \mathrm{ABC} ; \angle \mathrm{ACB}\) is a right angle. Prove: \(\mathrm{AD} / \mathrm{DB}=(\mathrm{AE})^{2} /(\mathrm{EB})^{2}\)

Chapter 14

The sides of a triangle have lengths 15,20 and \(28 .\) Find the lengths of the segment into which the bisector of the angle with the greatest measure divides the opposite side.

Chapter 14

In an isosceles trapezoid, the length of the lower base is 15 , the length of the upper base is 5, and each congruent side is of length 6 (see figure). By how many units must each nonparallel side be extended to form a triangle?

Chapter 14

The sides of triangle \(\mathrm{ABC}\) measure 5,7, and 9 . The shortest side of a similar triangle, $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$, measures 10 . (a) Find the measure of the longest side of triangle $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$. (b) Find the ratio of the measures of a pair of corresponding altitudes in triangles \(\mathrm{ABC}\) and $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$. (c) Find the perimeter of triangle $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$.

Chapter 14

A right triangle has legs of length 6 and 8 inches. CD bisects the right angle. Find the lengths of \(\underline{A D}\) and \(\underline{D B}\).

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