Question 3 - Area of Triangles - Areas of Parallelograms and Triangles

Last updated at April 16, 2024 by Teachoo

Since rectangle, square, rhombus are all parallelograms.

We can also say that

Diagonals of a rectangle divide it into 4 triangles of equal area

OR

Diagonals of a square divide it into 4 triangles of equal area

OR

Diagonals of a rhombus divide it into 4 triangles of equal area

Transcript

Question 3
Show that the diagonals of a parallelogram divide it into four triangles of equal area.
Given: A parallelogram ABCD
With diagonals AC & BD
To prove: ar (∆AOB) = ar (∆BOC) = ar (∆COD) = ar (∆AOD)
Proof : ABCD is a parallelogram
Diagonals of a parallelogram bisect each other
∴ O is the mid-point of BD, i.e., OB = OD
& O is the mid-point of AC, i.e., OA = OC
In ∆ ABC,
Since OA = OC
∴ BO is the median of ∆ ABC
⇒ ar(∆ AOB) = ar(∆ BOC)
In ∆ ADC,
Since OA = OC
∴ DO is the median of ∆ ADC
⇒ ar(∆ AOD) = ar(∆ COD)
Similarly,
In ∆ABD,
Since OB = OD
∴ AO is the median of ∆ ABD
⇒ ar(∆ AOB) = ar(∆ AOD)
From (3) , (4) & (5)
ar (∆AOB) = ar (∆BOC) = ar (∆COD) = ar (∆AOD)
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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