In figure, if \( \angle \mathrm{A}=\angle \mathrm{C}, \mathrm{AB}=6 \mathrm{~cm}, \mathrm{BP}=15 \mathrm{~cm} \), \( \mathrm{AP}=12 \mathrm{~cm} \) and \( \mathrm{CP}=4 \mathrm{~cm} \), then find the lengths of \( \mathrm{PD} \) and CD.
"
Given:
\( \angle \mathrm{A}=\angle \mathrm{C}, \mathrm{AB}=6 \mathrm{~cm}, \mathrm{BP}=15 \mathrm{~cm} \), \( \mathrm{AP}=12 \mathrm{~cm} \) and \( \mathrm{CP}=4 \mathrm{~cm} \).
To do:
We have to find the lengths of \( \mathrm{PD} \) and CD.
Solution:
In $\triangle \mathrm{APB}$ and $\triangle \mathrm{CPD}$,
$\angle \mathrm{A}=\angle \mathrm{C}$ (Given)
$\angle \mathrm{APS}=\angle \mathrm{CPD}$ (Vertically opposite angles)
Therefore, by AA similarity,
$\triangle A P D \sim \triangle C P D$
This implies,
$\frac{A P}{C P}=\frac{P B}{P D}=\frac{A B}{C D}$
$\frac{12}{4}=\frac{15}{P D}=\frac{6}{C D}$
Therefore,
$P D=\frac{15 \times 4}{12}$
$=5 \mathrm{~cm}$
$C D=\frac{6 \times 4}{12}$
$=2 \mathrm{~cm}$
Hence, the length of $P D$ is $5 \mathrm{~cm}$ and the length of $C D$ is $2 \mathrm{~cm}$.
Related Articles
- Find the area of a quadrilateral \( \mathrm{ABCD} \) in which \( \mathrm{AB}=3 \mathrm{~cm}, \mathrm{BC}=4 \mathrm{~cm}, \mathrm{CD}=4 \mathrm{~cm} \), \( \mathrm{DA}=5 \mathrm{~cm} \) and \( \mathrm{AC}=5 \mathrm{~cm} \).
- If \( \Delta \mathrm{ABC} \sim \Delta \mathrm{DEF}, \mathrm{AB}=4 \mathrm{~cm}, \mathrm{DE}=6 \mathrm{~cm}, \mathrm{EF}=9 \mathrm{~cm} \) and \( \mathrm{FD}=12 \mathrm{~cm} \), find the perimeter of \( \triangle \mathrm{ABC} \).
- In figure below, if \( \angle \mathrm{ACB}=\angle \mathrm{CDA}, \mathrm{AC}=8 \mathrm{~cm} \) and \( \mathrm{AD}=3 \mathrm{~cm} \), find \( \mathrm{BD} \)."
- Choose the correct answer from the given four options:It is given that \( \triangle \mathrm{ABC} \sim \triangle \mathrm{DFE}, \angle \mathrm{A}=30^{\circ}, \angle \mathrm{C}=50^{\circ}, \mathrm{AB}=5 \mathrm{~cm}, \mathrm{AC}=8 \mathrm{~cm} \) and \( D F=7.5 \mathrm{~cm} \). Then, the following is true:(A) \( \mathrm{DE}=12 \mathrm{~cm}, \angle \mathrm{F}=50^{\circ} \)(B) \( \mathrm{DE}=12 \mathrm{~cm}, \angle \mathrm{F}=100^{\circ} \)(C) \( \mathrm{EF}=12 \mathrm{~cm}, \angle \mathrm{D}=100^{\circ} \)(D) \( \mathrm{EF}=12 \mathrm{~cm}, \angle \mathrm{D}=30^{\circ} \)
- \( \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR} . \quad \) If \( \quad \mathrm{AB}+\mathrm{BC}=12 \mathrm{~cm} \) \( \mathrm{PQ}+\mathrm{QR}=15 \mathrm{~cm} \) and \( \mathrm{AC}=8 \mathrm{~cm} \), find \( \mathrm{PR} \).
- \( \triangle \mathrm{ABC} \sim \triangle \mathrm{ZYX} . \) If \( \mathrm{AB}=3 \mathrm{~cm}, \quad \mathrm{BC}=5 \mathrm{~cm} \), \( \mathrm{CA}=6 \mathrm{~cm} \) and \( \mathrm{XY}=6 \mathrm{~cm} \), find the perimeter of \( \Delta \mathrm{XYZ} \).
- Name the types of following triangles:(a) Triangle with lengths of sides \( 7 \mathrm{~cm}, 8 \mathrm{~cm} \) and \( 9 \mathrm{~cm} \).(b) \( \triangle \mathrm{ABC} \) with \( \mathrm{AB}=8.7 \mathrm{~cm}, \mathrm{AC}=7 \mathrm{~cm} \) and \( \mathrm{BC}=6 \mathrm{~cm} \).(c) \( \triangle \mathrm{PQR} \) such that \( \mathrm{PQ}=\mathrm{QR}=\mathrm{PR}=5 \mathrm{~cm} \).(d) \( \triangle \mathrm{DEF} \) with \( \mathrm{m} \angle \mathrm{D}=90^{\circ} \)(e) \( \triangle \mathrm{XYZ} \) with \( \mathrm{m} \angle \mathrm{Y}=90^{\circ} \) and \( \mathrm{XY}=\mathrm{YZ} \).(f) \( \Delta \mathrm{LMN} \) with \( \mathrm{m} \angle \mathrm{L}=30^{\circ}, \mathrm{m} \angle \mathrm{M}=70^{\circ} \) and \( \mathrm{m} \angle \mathrm{N}=80^{\circ} \).
- It is given that \( \triangle \mathrm{ABC} \sim \Delta \mathrm{EDF} \) such that \( \mathrm{AB}=5 \mathrm{~cm} \), \( \mathrm{AC}=7 \mathrm{~cm}, \mathrm{DF}=15 \mathrm{~cm} \) and \( \mathrm{DE}=12 \mathrm{~cm} \). Find the lengths of the remaining sides of the triangles.
- Construct a triangle \( \mathrm{ABC} \) in which \( \mathrm{BC}=7 \mathrm{~cm}, \angle \mathrm{B}=75^{\circ} \) and \( \mathrm{AB}+\mathrm{AC}=13 \mathrm{~cm} \).
- Construct a triangle \( \mathrm{ABC} \) in which \( \mathrm{BC}=8 \mathrm{~cm}, \angle \mathrm{B}=45^{\circ} \) and \( \mathrm{AB}-\mathrm{AC}=3.5 \mathrm{~cm} \).
- Construct a triangle \( \mathrm{PQR} \) in which \( \mathrm{QR}=6 \mathrm{~cm}, \angle \mathrm{Q}=60^{\circ} \) and \( \mathrm{PR}-\mathrm{PQ}=2 \mathrm{~cm} \).
- Choose the correct answer from the given four options:If \( \triangle \mathrm{ABC} \sim \Delta \mathrm{QRP}, \frac{\operatorname{ar}(\mathrm{ABC})}{\operatorname{ar}(\mathrm{PQR})}=\frac{9}{4}, \mathrm{AB}=18 \mathrm{~cm} \) and \( \mathrm{BC}=15 \mathrm{~cm} \), then \( \mathrm{PR} \) is equal to(A) \( 10 \mathrm{~cm} \)(B) \( 12 \mathrm{~cm} \)(C) \( \frac{20}{3} \mathrm{~cm} \)(D) \( 8 \mathrm{~cm} \)
- In \( \Delta \mathrm{XYZ}, \mathrm{S} \) and \( \mathrm{T} \) are points of \( \mathrm{XY} \) and \( \mathrm{XZ} \) respectively and ST \( \| \mathrm{YZ} \). If \( \mathrm{XS}=4 \mathrm{~cm} \), \( \mathrm{XT}=8 \mathrm{~cm}, \mathrm{SY}=x-4 \mathrm{~cm} \) and \( \mathrm{TZ}=3 x-19 \mathrm{~cm} \) find the value of \( x \).
Kickstart Your Career
Get certified by completing the course
Get Started