ICSE Class 10 Answered
pls answer this question
Asked by shethdhanesh | 30 Nov, 2023, 11:35: PM
Expert Answer
PL:LQ = 2 :3 .................................(1)
Since LM || QR , we get
PM: MR = 2:3 ............................(2)
Since LM || QR , 
LPM = NMR ...............(3)
LPM = NMR ...............(3)From above equations (1), (2) and (3) and by SAS theorem of similar triangles , we get
ΔPLM ~ ΔMNR
-----------------------------------------------------
Let us draw perpendicular lines PS and MT.
Since PS || MT , ΔMTR and ΔPSR are similar
PS/MT = PR/MR = (PM+MR) / MR = (PM/MR)+ 1 = (2/3)+1
In above expression, eqn.(2) is used to substitute (PM/MR) .
Hence , PS/MT = PR/ MR = 5/3 ................................(4)
PS/MT = (PU+US)/MT = (PU/MT) + 1 ( because US=MT )
Hence , we get
PS/MT = (PU/MT) +1 = 5/3
PU/MT = (5/3)-1 = 2/3 .................................(5)
Since LM || QR , ΔPLM and ΔPQR are similar triangles
PQ/PL = (PL+LQ)/PL = 1 +(LQ/PL) = 1 + (3/2) = 5/2
Hence QR : LM = 5 : 2 .................................(6)
QR/LM = ( QN+NR ) / LM = ( QN/LM ) + ( NR/LM ) = 1 + ( NR/LM ) = 5/2 ( because QN = LM )
Hence NR/LM = 3/2 ..........................(7)
Area of ΔPLM : Area of ΔMNR = (1/2) (LM×PU) : (1/2) (NR × MT )
Area of ΔPLM : Area of ΔMNR = (1/2) (2×2) : (1/2) (3 × 3 ) ( using eqn. (5) and eqn.(7) )
Area of ΔPLM : Area of ΔMNR = 4 : 9
------------------------------------------------------------------------------------
Area of ΔPLM : Area of ΔPQR = (1/2) (LM×PU) : (1/2) (QR × PS )
PS/PU = (PU+US)/PU = 1 + (US/PU) = 1 +(MT/PU) = 1+(3/2) = ( 5/2) [ eqn. (5) is used for (MT/PU) ]
LM/QR = 2/5 ( refer eqn.(6) )
Area of ΔPLM : Area of ΔPQR = (1/2) (2×2) : (1/2) (5 × 5 ) = 4 : 25
------------------------------------------------------------------------------
Since ΔMNR and ΔPQR are similar triangles ,
QR/NR = PR/MR
QR / NR = ( PM+MR )/ MR
QR/NR = 1+ ( PM/MR) = 1+ (2/3) = 5/3
From eqn.(4) , we have PS/MT = 5.3
Area of ΔMNR : Area of ΔPQR = (1/2) (3×3) : (1/2) (5 × 5 ) = 9 : 5
Answered by Thiyagarajan K | 01 Dec, 2023, 03:20: PM
Application Videos
-
Arithmetic Progression - 1
This video explains the concept of Arithmetic Progression, Solving Linear E... -
Height and distance_Application of trigo 1 ...
This video explains an application based on heights and distances. -
Height and distance_Application of trigo 2 ...
This video explains an application based on heights and distances. -
Height and distance_Application of trigo 3 ...
This video explains an application based on heights and distances. -
Finding probability_Probability 1
This video explains an application based on finding probability
Concept Videos
-
Remainder Theorem
Discuss various algebraic operations on Polynomials. Learn the statement a... -
Factorisation: Factor Theorem
This video explains the factorization of polynomials using Factor Theorem a... -
Introduction to Quadratic Equations ...
This video explains the definition of quadratic equation and solve the rela... -
Method of Completing the Square ...
This video explains the determination of roots of the quadratic equation by... -
Introduction to Matrices
This video explains what a matrix is, its order and discusses the types of ...
ICSE 10 - Maths
Asked by bhagvantingre | 24 Apr, 2024, 03:05: PM
ANSWERED BY EXPERTICSE 10 - Maths
Asked by shaikafridi860 | 21 Apr, 2024, 11:22: PM
ANSWERED BY EXPERTICSE 10 - Maths
Asked by aritramalakar4 | 13 Apr, 2024, 08:33: PM
ANSWERED BY EXPERTICSE 10 - Maths
Asked by ashwanikaler9 | 11 Apr, 2024, 01:50: PM
ANSWERED BY EXPERT
ICSE 10 - Maths
Asked by sagarmishra | 15 Mar, 2024, 03:22: PM
ANSWERED BY EXPERTICSE 10 - Maths
Asked by jesnamariaroy2626 | 10 Mar, 2024, 11:28: AM
ANSWERED BY EXPERTICSE 10 - Maths
Asked by vrj.parmar369 | 07 Mar, 2024, 11:27: AM
ANSWERED BY EXPERT
