$ \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} $.
Given:
\( \Delta \mathrm{ABC} \sim \Delta \mathrm{XZY} \).
The perimeter of \( \triangle \mathrm{ABC} \) is \( 45 \mathrm{~cm} \), the perimeter of \( \triangle \mathrm{XYZ} \) is \( 30 \mathrm{~cm} \) and \( \mathrm{AB}=21 \mathrm{~cm} \).
To do:
We have to find \( \mathrm{XZ} \).
Solution:
\( \triangle \mathrm{ABC} \sim \triangle \mathrm{XYZ} \)
The ratio of the perimeters of two similar triangles is same as the ratio of their corresponding sides.
Therefore,
$\frac{Perimeter\ of\ \triangle ABC}{Primeter\ of\ \triangle XYZ}=\frac{AB}{XY}$
This implies,
$\frac{45}{30}=\frac{21}{XY}$
$XY=\frac{21\times30}{45}$
$XY=14\ cm$
Hence, \( \mathrm{XZ}=14\ cm \).
Related Articles
- \( \triangle \mathrm{ABC} \sim \triangle \mathrm{ZYX} . \) 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 \( \Delta \mathrm{XYZ} \).
- 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 \( \triangle \mathrm{ABC} \).
- \( \triangle \mathrm{ABC} \sim \triangle \mathrm{EFD} \). 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 \( \triangle \mathrm{DEF} \).
- \( \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR} . \quad \) If \( \quad \mathrm{AB}+\mathrm{BC}=12 \mathrm{~cm} \) \( \mathrm{PQ}+\mathrm{QR}=15 \mathrm{~cm} \) and \( \mathrm{AC}=8 \mathrm{~cm} \), find \( \mathrm{PR} \).
- 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 \( \triangle \mathrm{ABC}, \angle \mathrm{A}=90^{\circ} \) and \( \mathrm{AM} \) is an altitude. If \( \mathrm{BM}=6 \) and \( \mathrm{CM}=2 \), find the perimeter of \( \triangle \mathrm{ABC} \).
- In \( \triangle \mathrm{ABC}, \angle \mathrm{B}=90^{\circ} \) and \( \mathrm{BM} \) is an altitude. If \( \mathrm{BM}=10 \) and \( \mathrm{CM}=5 \), find the perimeter of \( \triangle \mathrm{ABC} \).
- \( \mathrm{X}, \mathrm{Y} \) and \( \mathrm{Z} \) are the midpoints of the sides of \( \Delta \mathrm{PQR} . \mathrm{A}, \mathrm{B} \) and \( \mathrm{C} \) are the midpoints of the sides of \( \triangle \mathrm{XYZ} \). If \( \mathrm{PQR}=240 \mathrm{~cm}^{2} \), find \( \mathrm{XYZ} \) and \( \mathrm{ABC} \).
- 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) \( 8 \mathrm{~cm} \)
- 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 \( \triangle \mathrm{ABC}, \mathrm{M} \) and \( \mathrm{N} \) are the midpoints of \( \mathrm{AB} \) and \( \mathrm{AC} \) respectively. If the area of \( \triangle \mathrm{ABC} \) is \( 90 \mathrm{~cm}^{2} \), find the area of \( \triangle \mathrm{AMN} \).
- 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 \( \mathrm{m} \angle \mathrm{N}=80^{\circ} \).
- 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 \( x \).
- Construct a triangle \( \mathrm{ABC} \) in which \( \mathrm{BC}=7 \mathrm{~cm}, \angle \mathrm{B}=75^{\circ} \) and \( \mathrm{AB}+\mathrm{AC}=13 \mathrm{~cm} \).
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