7. A farmer was having a field in the form of a parallelogram PQRS. She took any point A on RS and joined it to points P and Q. In how many parts the fields is divided? What are the shapes of these parts? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it?
Solution:
The field is divided into three parts each in triangular shape.
Let, ΔPSA, ΔPAQ and ΔQAR be the triangles.
Area of (ΔPSA + ΔPAQ + ΔQAR) = Area of PQRS — (i)
Area of ΔPAQ = ½ area of PQRS — (ii)
Here, the triangle and parallelogram are on the same base and in-between the same parallel lines.
From (i) and (ii),
Area of ΔPSA +Area of ΔQAR = ½ area of PQRS — (iii)
From (ii) and (iii), we can conclude that,
The farmer must sow wheat or pulses in ΔPAQ or either in both ΔPSA and ΔQAR.
8. In Fig, E is any point on median AD of a ΔABC. Show that ar (ABE) = ar(ACE).
Solution:
Given,
AD is median of ΔABC.
∴ it will divide ΔABC into two triangles of equal area.
∴ ar(ABD) = ar(ACD) — (i)
also,
ED is the median of ΔABC.
∴ ar(EBD) = ar(ECD) — (ii)
Subtracting (ii) from (i),
ar(ABD) – ar(EBD) = ar(ACD) – ar(ECD)
⇒ ar(ABE) = ar(ACE)
9. In a triangle ABC, E is the mid-point of median AD. Show that ar(BED) = ¼ ar(ABC).
Solution:
ar(BED) = (1/2) × BD × DE
Since, E is the mid-point of AD,
AE = DE
Since, AD is the median on side BC of triangle ABC,
BD = DC
DE = (1/2) AD — (i)
BD = (1/2)BC — (ii)
From (i) and (ii), we get,
ar(BED) = (1/2) × (1/2)BC × (1/2)AD
⇒ ar(BED) = (1/2) × (1/2) ar(ABC)
⇒ ar(BED) = ¼ ar(ABC)
10. Show that the diagonals of a parallelogram divide it into four triangles of equal area.
Solution:
O is the mid point of AC and BD. (diagonals of bisect each other)
In ΔABC, BO is the median.
∴ ar(AOB) = ar(BOC) — (i)
also,
In ΔBCD, CO is the median.
∴ar(BOC) = ar(COD) — (ii)
In ΔACD, OD is the median.
∴ ar(AOD) = ar(COD) — (iii)
In ΔABD, AO is the median.
∴ar(AOD) = ar(AOB) — (iv)
From equations (i), (ii), (iii) and (iv), we get,
ar(BOC) = ar(COD) = ar(AOD) = ar(AOB)
Hence, we get, the diagonals of a parallelogram divide it into four triangles of equal area.
11. In Fig, ABC and ABD are two triangles on the same base AB. If line- segment CD is bisected by AB at O, show that: ar(ABC) = ar(ABD).
Solution:
In ΔABC, AO is the median. (CD is bisected by AB at O)
∴ ar(AOC) = ar(AOD) — (i)
also,
ΔBCD, BO is the median. (CD is bisected by AB at O)
∴ar(BOC) = ar(BOD) — (ii)
Adding (i) and (ii),
We get,
ar(AOC) + ar(BOC) = ar(AOD) + ar(BOD)
⇒ ar(ABC) = ar(ABD)
12. D, E and F are respectively the mid-points of the sides BC, CA and AB of a ΔABC.
Show that
(i) BDEF is a parallelogram.
(ii) ar(DEF) = ¼ ar(ABC)
(iii) ar (BDEF) = ½ ar(ABC)
Solution:
(i) In ΔABC,
EF || BC and EF = ½ BC (by mid point theorem)
also,
BD = ½ BC (D is the mid point)
So, BD = EF
also,
BF and DE are parallel and equal to each other.
∴ the pair opposite sides are equal in length and parallel to each other.
∴ BDEF is a parallelogram.
(ii) Proceeding from the result of (i),
BDEF, DCEF, AFDE are parallelograms.
Diagonal of a parallelogram divides it into two triangles of equal area.
∴ ar(ΔBFD) = ar(ΔDEF) (For parallelogram BDEF) — (i)
also,
ar(ΔAFE) = ar(ΔDEF) (For parallelogram DCEF) — (ii)
ar(ΔCDE) = ar(ΔDEF) (For parallelogram AFDE) — (iii)
From (i), (ii) and (iii)
ar(ΔBFD) = ar(ΔAFE) = ar(ΔCDE) = ar(ΔDEF)
⇒ ar(ΔBFD) + ar(ΔAFE) + ar(ΔCDE) + ar(ΔDEF) = ar(ΔABC)
⇒ 4 ar(ΔDEF) = ar(ΔABC)
⇒ ar(DEF) = ¼ ar(ABC)
(iii) Area (parallelogram BDEF) = ar(ΔDEF) + ar(ΔBDE)
⇒ ar(parallelogram BDEF) = ar(ΔDEF) + ar(ΔDEF)
⇒ ar(parallelogram BDEF) = 2 × ar(ΔDEF)
⇒ ar(parallelogram BDEF) = 2 × ¼ ar(ΔABC)
⇒ ar(parallelogram BDEF) = ½ ar(ΔABC)