Question 33 - CBSE Class 10 Sample Paper for 2023 Boards - Maths Standard - Solutions of Sample Papers for Class 10 Boards

Last updated at May 29, 2023 by Teachoo

Prove that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.

Using the above theorem prove that a line through the point of intersection of the diagonals and parallel to the base of the trapezium divides the non parallel sides in the same ratio.

Question 33 Prove that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. Using the above theorem prove that a line through the point of intersection of the diagonals and parallel to the base of the trapezium divides the non parallel sides in the same ratio.Let’s prove the theorem first
Theorem If a line is drawn parallel to one side of a triangle to intersect the other two side in distinct points, the other two sides are divided in the same ratio. Given: Δ ABC where DE ∥ BC
To Prove: 𝐴𝐷/𝐷𝐵 = 𝐴𝐸/𝐸𝐶
Construction: Join BE and CD
Draw DM ⊥ AC and EN ⊥ AB.
Proof:ar (ADE) = 1/2 × Base × Height
= 1/2 × AD × EN
ar (BDE) = 1/2 × Base × Height
= 1/2 × DB × EN
Divide (1) and (2)
"ar (ADE)" /"ar (BDE)" = (1/2 " × AD × EN" )/(1/2 " × DB × EN " )
"ar (ADE)" /"ar (BDE)" = "AD" /"DB"
ar (ADE) = 1/2 × Base × Height
= 1/2 × AE × DM
ar (DEC) = 1/2 × Base × Height
= 1/2 × EC × DM
Divide (3) and (4)
"ar (ADE)" /"ar (DEC)" = (1/2 " × AE × DM" )/(1/2 " × EC × DM " )
"ar (ADE)" /"ar (DEC)" = "AE" /"EC"
Now,
∆BDE and ∆DEC are on the same base DE
and between the same parallel lines BC and DE.
∴ ar (BDE) = ar (DEC)
Hence,
"ar (ADE)" /"ar (BDE)" = "ar (ADE)" /"ar (DEC)"
"AD" /"DB" = "AE" /"EC"
Hence Proved
Now, let’s look at our questionUsing the above theorem prove that a line through the point of intersection of the diagonals and parallel to the base of the trapezium divides the non parallel sides in the same ratio. Given: ABCD is a trapezium
where diagonals AC & BD intersect at O
EF is a line passing through O, parallel to CD
To prove: We need to prove 𝑨𝑬/𝑬𝑫=𝑩𝑭/𝑭𝑪
Proof:
Using theorem If a line is drawn parallel to one side of a triangle to intersect the other two side in distinct points, the other two sides are divided in the same ratio
In Δ ADB
Since EO ∥ CD
Using Basic Proportionality theorem
𝑨𝑬/𝑫𝑬=𝑩𝑶/𝑫𝑶
In Δ BDC
Since OF ∥ CD
Using Basic Proportionality theorem
𝑩𝑶/𝑫𝑶=𝑩𝑭/𝑭𝑪
Comparing (1) and (2)
𝑨𝑬/𝑬𝑫=𝑩𝑭/𝑭𝑪
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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