$ \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} $.
Given:
\( \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR} .\)
\( \mathrm{AB}+\mathrm{BC}=12 \mathrm{~cm} \) \( \mathrm{PQ}+\mathrm{QR}=15 \mathrm{~cm} \) and \( \mathrm{AC}=8 \mathrm{~cm} \).
To do:
We have to find \( \mathrm{PR} \).
Solution:
\( \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR} \)
The ratio of the perimeters of two similar triangles is same as the ratio of their corresponding sides.
Therefore,
$\frac{Perimeter\ of\ \triangle ABC}{Primeter\ of\ \triangle PQR}=\frac{AC}{PR}$
This implies,
$\frac{AB+BC+CA}{PQ+QR+RP}=\frac{8}{PR}$
$PR(12+8)=8(15+PR)$
$20PR=120+8PR$
$(20-8)PR=120$
$12PR=120$
$PR=10\ cm$
Hence, \( \mathrm{PR}=10\ cm \).
- Related Articles
- \( \triangle \mathrm{PQR} \sim \triangle \mathrm{ZYX} . \quad \) If \( \mathrm{PQ}: \mathrm{ZY}=5: 3 \) and \( \mathrm{PR}=10 \mathrm{~cm} \), find \( \mathrm{ZX} \).
- \( \triangle \mathrm{ABC} \sim \triangle \mathrm{ZYX} . \) 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 \( \Delta \mathrm{XYZ} \).
- Choose the correct answer from the given four options:If in two triangles \( \mathrm{ABC} \) and \( \mathrm{PQR}, \frac{\mathrm{AB}}{\mathrm{QR}}=\frac{\mathrm{BC}}{\mathrm{PR}}=\frac{\mathrm{CA}}{\mathrm{PQ}} \), then(A) \( \triangle \mathrm{PQR} \sim \triangle \mathrm{CAB} \)(B) \( \triangle \mathrm{PQR} \sim \triangle \mathrm{ABC} \)(C) \( \triangle \mathrm{CBA} \sim \triangle \mathrm{PQR} \)(D) \( \triangle \mathrm{BCA} \sim \triangle \mathrm{PQR} \)
- 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 \( \triangle \mathrm{ABC} \).
- Construct a triangle \( \mathrm{PQR} \) in which \( \mathrm{QR}=6 \mathrm{~cm}, \angle \mathrm{Q}=60^{\circ} \) and \( \mathrm{PR}-\mathrm{PQ}=2 \mathrm{~cm} \).
- Construct a triangle \( \mathrm{ABC} \) in which \( \mathrm{BC}=8 \mathrm{~cm}, \angle \mathrm{B}=45^{\circ} \) and \( \mathrm{AB}-\mathrm{AC}=3.5 \mathrm{~cm} \).
- In \( \triangle \mathrm{ABC}, \mathrm{AD} \) is a median. If \( \mathrm{AB}=18 \mathrm{~cm} \). \( \mathrm{AC}=14 \mathrm{~cm} \) and \( \mathrm{AD}=14 \mathrm{~cm} \), find the perimeter of \( \triangle \mathrm{ABC} \).
- 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 \( \triangle \mathrm{PQR}, \quad \angle \mathrm{P}=\angle \mathrm{Q}+\angle \mathrm{R}, \mathrm{PQ}=7 \) and \( \mathrm{QR}=25 \). Find the perimeter of \( \triangle \mathrm{PQR} \).
- Construct a triangle \( \mathrm{ABC} \) in which \( \mathrm{BC}=7 \mathrm{~cm}, \angle \mathrm{B}=75^{\circ} \) and \( \mathrm{AB}+\mathrm{AC}=13 \mathrm{~cm} \).
- \( \Delta \mathrm{ABC} \sim \Delta \mathrm{XZY} \). If the perimeter of \( \triangle \mathrm{ABC} \) is \( 45 \mathrm{~cm} \), the perimeter of \( \triangle \mathrm{XYZ} \) is \( 30 \mathrm{~cm} \) and \( \mathrm{AB}=21 \mathrm{~cm} \), find \( \mathrm{XY} \).
- 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.
- In \( \triangle \mathrm{ABC}, \mathrm{AD} \) is a median. If \( \mathrm{AB}=8, \mathrm{AC}=15 \) and \( \mathrm{AD}=8.5 \), find \( \mathrm{BC} \).
- \( \triangle \mathrm{ABC} \sim \triangle \mathrm{EFD} \). If \( \mathrm{AB}: \mathrm{BC}: \mathrm{CA}=4: 3: 5 \) and the perimeter of \( \triangle \mathrm{DEF} \) is \( 36 \mathrm{~cm} \), find all the sides of \( \triangle \mathrm{DEF} \).
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies