Parallelograms

Exercise 9.4

Part 1

Question 1: Parallelogram ABCD and rectangle ABEF are on the same base AB and have equal areas. Show that the perimeter of the parallelogram is greater than that of the rectangle.

Answer: In this figure, ABCD is a parallelogram and EFCD is a rectangle. They are on the same base DC.

parallelogram

Height of parallelogram = FC

AB = DC (Opposite sides of parallelogram)

EF = DC (Opposite sides of rectangle)

From above two equations, it is clear that

EF = DC

This means EA = FB

Perimeter of ABCD = AB + BC + CD + AD

Perimeter of EFCD = EF + FC + CD + ED

= AB + CD + FC + ED

In ΔEAD

AD > ED (hypotenuse is the longest side)

In ΔFBC

BC > FC (hypotenuse is the longest side)

So, AB + CD + BC + AD > AB + CD + FC + ED

So, it is proved that perimeter of parallelogram is greater than perimeter of rectangle made on the same base.


Question 2: In this figure, D and E are two points on BC such that BD = DE = EC. Show that ar(ABD) = ar(ADE) = ar(AEC).

triangle

Can you now answer the question that you have left in the ‘Introduction’ of this chapter, whether the field of Budhia has been actually divided into three parts of equal area?

Answer: ΔABD, ΔADE and ΔAEC all have same height = h

BD = DE = EC (Given)

So, bases are equal

So, area of anyone of the three triangles = `1/2xx\h\xxb`

So, ar(ABD) = ar(ADE) = ar(AEC) PROVED

Question 3: In this figure, ABCD, DCFE and ABFE are parallelograms. Show that ar(ADE) = ar(BCF)

parallelogram

Answer: In ΔADE and Δ BCF

AE = BF (Opposite sides of parallelogram ABFE)

AD = BC (Opposite sides of parallelogram ABCD)

DE = CF (Opposite sides of parallelogram DCFE)

So, from SSS theorem

ΔADE ≅ Δ BCF

Or, ar(ΔADE) = ar(Δ BCF) PROVED

Question 4: In this figure, ABCD is a parallelogram and BC is produced to a point Q such that AD = CQ. If AQ intersects DC at P, show that ar(BPC) = ar(DPQ). (Hint: Join AC)

parallelogram

Answer: AD = CQ (given)

AD = BC (Opposite sides of parallelogram ABCD)

So, AD = CQ = BC

ar(ΔQAC) = ar(ΔQDC)

(Triangles on same base QC and between same parallels DA and QC)

Subtracting ΔQPC from both sides:

ar(ΔQAC - ΔQPC) = ar(ΔQDC - ΔQPC)

= ar(ΔAPC) = ar(ΔDPQ) ………..(1)

Now, ar(ΔPAC) = ar(ΔPBC) ………..(2)

(Triangles on the same base PC and between same parallels AB and PC)

From equations (1) and (2)

ar(ΔBPC) = ar(ΔDPQ) PROVED


Question 5: In this figure, ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that

(Hint: Join EC and AD. Show that BE||AC and DE||AB, etc.)

triangle

(a) ar(BDF) = `1/4` ar(ABC)

Answer: Side of ΔABC = s

Side of ΔBDE = `s/2` (Because BD = DC)

ar(ABC) `=(sqrt3)/4xxs^2`

ar(BDE) `=(sqrt2)/4xx(s/2)^2=(sqrt3)/4xx(s^2)/4`

`(BD\E)/(AB\C)=1/4`

Or, ar(BDE) = `1/4`ar(ABC) proved

(b) ar(BDE) = `1/2` ar(BAE)

Answer: ∠ACB = 60°

∠DBE = 60°

(angles of equilateral triangles)

So, AC||BE

So, ar(ΔBAE) = ar(BEC)

(Triangles on same base BE and between same parallels AC and BE)

Since BD = `1/2` BC

So, ar(ΔBDE) = `1/2` ar(ΔBEC)

Or, ar(ΔBDE) = `1/2` ar(ΔBAE) PROVED

(c) ar(ABC) = 2 ar(BEC)

Answer: Since BD = `1/2`BC = `1/2`AC

So, PE = `1/2`AD

So, ar(ABC) = `1/2` AD × BC

ar(BEC) = `1/2xx1/2` AD × BC

Or, `(ar(BE\C))/(ar(AB\C))=1/2`

Or, ar(ABC) = 2 ar(BEC) PROVED

(d) ar(BFE) = ar(AFD)

Answer: ΔABC and ΔBDE are equilateral triangles.

So,∠ABC = &BDE = 60°

So, AB||DE

ΔBED and ΔAED are on the same base ED and between same parallels AB and DE

So, ar(ΔBED) = ar(ΔAED)

Subtracting ar(ΔEFD) from both sides,

Ar(ΔBED) – ar(ΔEFD) = ar(ΔAED) – ar(ΔEFD)

= ar(ΔBFE) = ar(ΔAFD) PROVED

(e) ar(BFE) = 2 ar(FED)

Answer: In right angle ΔABD

AD2 = AB2 - BD2

Or, AD2 `a^2-(a^2)/4`

`=(4a^2-a^2)/4=(3a^2)/4`

Or, AD `=(sqrt3a)/2`

In right angle ΔPED

EP2 = DE2 - DP2

Or, EP2 `=(a/2)^2-(a/4)^2`

`=(a^2)/4-(a^2)/(16)=(3a^2)/(16)`

Or, EP `=(sqrt3a)/4`

Now, ar(ΔAFD) = `1/2` × FD × AD

`=1/2`× FD × `(sqrt3)/2`a ……..(1)

Similarly, ar(ΔEFD) `=1/2` × FD × EP

`=1/2` × FD × `(sqrt3)/4`a …………….(2)

From equations (1) and (2)

Ar(ΔAFD) = 2 ar(ΔEFD)

From answer to question (d)

Ar(ΔAFD) = ar(ΔBEF)

Or, ar(ΔBFE) = 2 ar(ΔEFD)

(f) ar(FED) = `1/8` ar(AFC)

Answer: ar(ΔAFC) = ar(ΔAFD) + ar(ΔADC)

= ar(ΔBFE) + `1/2` ar(ΔABC) (From question (d) )

= ar(ΔBFE) + `1/2` × 4 × ar(ΔBDE) (From question (a) )

= ar(ΔBFE) + 2 ar(ΔBDE)

= 2 ar(ΔFED) + 2(ar(ΔBFE) + ar(ΔFED))

= 2 ar(ΔFED) + 2(2 ar(ΔFED) + ar(ΔFED)) (From question (e))

= 2ar(ΔFED) + 2(3 ar(ΔFED))

= 2 ar(ΔFED) + 6 ar(ΔFED)

= 8 ar(ΔFED)

Or, ar(ΔFED) = `1/8` ar(ΔAFC) PROVED



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