## Chapter 21 - Areas Theorems on Parallelograms Exercise Ex. 21.1

Question 1
In the given figure, if AB ∥ DC ∥ FG and AE is a straight line. Also, AD ∥ FC. Prove that: area of ∥ gm ABCD = area of ∥ gm BFGE.

Solution 1
Joining AC and FE, we get
ΔAFC and ΔAFE are on the same base AF and between the same parallels AF and CE.
A(ΔAFC) = A(ΔAFE)
A(ΔABF) + A(ΔABC) = A(ΔABF) + A(ΔBFE)
A(ΔABC) = A(ΔBFE)
A(parallelogram ABCD) = A(parallelogram BFGE) ⇒ A(parallelogram ABCD) = A(parallelogram BFGE)
Question 2
In the given figure, the perimeter of parallelogram PQRS is 42 cm. Find the lengths of PQ and PS.
Solution 2
Question 3
In the given figure, PT ∥ QR and QT ∥ RS. Show that: area of ΔPQR = area of ΔTQS.

*Question modified
Solution 3
Joining TR, we get
ΔPQR and ΔQTR are on the same base QR and between the same parallel lines QR and PT.
ΔQTR and ΔTQS are on the same base QT and between the same parallel lines QT and RS.
From (i) and (ii), we get
Question 4
In the given figure, ΔPQR is right-angled at P. PABQ and QRST are squares on the side PQ and hypotenuse QR. If PN ⊥ TS, show that:
(a) ΔQRB ≅ ΔPQT
(b) Area of square PABQ = area of rectangle QTNM.
Solution 4
Question 5
The diagonals of a parallelogram ABCD intersect at O. A line through O meets AB in P and CD in Q. Show that

Solution 5
Question 6
In the given figure AF = BF and DCBF is a parallelogram. If the area of ΔABC is 30 square units, find the area of the parallelogram DCBF.
Solution 6
Question 7
In the given figure, BC ∥ DE.
(a) If area of ΔADC is 20 sq. units, find the area of ΔAEB.
(b) If the area of ΔBFD is 8 square units, find the area of ΔCEF
Solution 7
Question 8
In the given figure, AB ∥ SQ ∥ DC and AD ∥ PR ∥ BC. If the area of quadrilateral ABCD is 24 square units, find the area of quadrilateral PQRS.
Solution 8

Question 9
In the given figure, PQ ∥ SR ∥ MN, PS ∥ QM and SM ∥ PN. Prove that:
ar. (SMNT) = ar. (PQRS).
Solution 9
SM ∥ PN
SM ∥ TN
Also, SR ∥ MN
ST ∥ MN
Hence, SMNT is a parallelogram.

SM ∥ PN
SM ∥ PO
Also, PS ∥ QM
PS ∥ OM
Hence, SMOP is a parallelogram.

Now, parallelograms SMNT and SMOP are on the same base SM and between the same parallels SM and PN.
A(parallelogram SMNT) = A(parallelogram SMOP) ….(i)

Similarly, we can show that quadrilaterals PQRS is a parallelogram.
Now, parallelograms PQRS and SMOP are on the same base PS and between the same parallels PS and QM.
A(parallelogram PQRS) = A(parallelogram SMOP) ….(ii)

From (i) and (ii), we have
A(parallelogram SMNT) = A(parallelogram PQRS)
Question 10
In the given figure, ABC is a triangle and AD is the median.
If E is any point on the median AD. Show that:
Area of ΔABE = Area of ΔACE.

Solution 10
AD is the median of ΔABC.
Therefore it will divide ΔABC into two triangles of equal areas.
Area(ΔABD) = Area(ΔACD) ….(i)

Similarly, ED is the median of ΔEBC.
Area(DEBD) = Area(DECD) ….(ii)

Subtracting equation (ii) from (i), we have
Area(ΔABD) - Area(ΔEBD) = Area(ΔACD) - Area(ΔECD)
Area(ΔABE) = Area(ΔACE)
Question 11
In the given figure, ABC is a triangle and AD is the median.
If E is the midpoint of the median AD, prove that:
Area of ΔABC = 4 × Area of ΔABE
Solution 11
AD is the median of ΔABC.
Therefore it will divide ΔABC into two triangles of equal areas.
Area(ΔABD) = Area(ΔACD) ….(i)

Similarly, ED is the median of ΔEBC.
Area(ΔEBD) = Area(ΔECD) ….(ii)

Subtracting equation (ii) from (i), we have
Area(ΔABD) - Area(ΔEBD) = Area(ΔACD) - Area(ΔECD)
⇒ Area(ΔABE) = Area(ΔACE) ….(iii)

Since E is the mid-point of median AD,
AE = ED

Now,
ΔABE and ΔBED have equal bases and a common vertex B.
∴ Area(ΔABE) = Area(ΔBED) ….(iv)

From (i), (ii), (iii) and (iv), we get
Area(ΔABE) = A(ΔBED) = Area(ΔACE) = Area(ΔEDC) ….(v)
Now,
Area(ΔABC) = Area(ΔABE) + A(ΔBED) + Area(ΔACE) + Area(ΔEDC)
= 4 × Area(ΔABE) [From (v)]
Question 12
In a parallelogram PQRS, M and N are the midpoints of the sides PQ and PS respectively. If area of ΔPMN is 20 square units, find the area of the parallelogram PQRS.
Solution 12
Construction: Join SM and SQ.
In a parallelogram PQRS, SQ is the diagonal.
So, it bisects the parallelogram.
∴ Area(DPSQ)

SM is the median of ΔPSQ.
∴ Area(ΔPSM)

Again, MN is the median of ΔPSM.
∴ Area(ΔPMN)

Question 13
In a parallelogram PQRS, T is any point on the diagonal PR. If the area of ΔPTQ is 18 square units find the area of ΔPTS.
Solution 13
Construction: Join QR. Let the diagonals PR and QS intersect each other at point O.
Since diagonals of a parallelogram bisect each other, therefore O is the mid-point of both PR and QS.
Now, median of a triangle divides it into two triangles of equal area.

In ΔPSQ, OP is the median.
Area(ΔPOS) = Area(ΔPOQ) ….(i)

Similarly, OT is the median of ΔTSQ.
Area(ΔTOS) = Area(ΔTOQ) ….(ii)

Subtracting equation (ii) from (i), we have
Area(ΔPOS) - Area(ΔTOS) = Area(ΔPOQ) - Area(ΔTOQ)
⇒ Area(ΔPTQ) = Area(ΔPTS)
⇒ Area(ΔPTS) = 18 square units
Question 14
In the given figure area of ∥ gm PQRS is 30 cm2. Find the height of ∥ gm PQFE if PQ = 6 cm.
Solution 14
Question 15
In the given figure, PQRS is a ∥ gm. A straight line through P cuts SR at point T and QR produced at N. Prove that area of triangle QTR is equal to the area of triangle STN.
Solution 15
ΔPQT and parallelogram PQRS are on the same base PQ and between the same parallel lines PQ and SR.
ΔPSN and parallelogram PQRS are on the same base PS and between the same parallel lines PS and QN.
Adding equations (i) and (ii), we get
Subtracting A(ΔRTN) from both the sides, we get
Question 16
In the given figure, ST ∥ PR. Prove that: area of quadrilateral PQRS = area of ΔPQT.
Solution 16
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Question 18
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Question 19
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Question 20
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Question 21
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Question 22
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Question 24
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Question 37
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