Chapter 15: Areas of Parallelograms And Triangles Exercise – 15.2
Question: 1
If figure, ABCD is a parallelogram, AE ⊥ DC and CF ⊥ AD. If AB = 16 cm, AE = 8 cm, and CF = 10 cm, Find AD.
Solution:
Given that,
In parallelogram ABCD, CD = AB = 16 cm [∵ Opposite side of a parallelogram are equal]
We know that,
Area of parallelogram = Base × Corresponding altitude
Area of parallelogram ABCD = CD × AE = AD × CF
16 cm × cm = AD × 10 cm
Thus, The length of AD is 12.8 cm.
Question: 2
In Q1, if AD = 6 cm, CF = 10 cm, and AE = 8 cm, Find AB.
Solution:
We know that,
Area of a parallelogram ABCD = AD × CF ⋅⋅⋅⋅⋅⋅ (1)
Again area of parallelogram ABCD = CD × AE⋅⋅⋅⋅⋅⋅ (2)
Compare equation (1) and equation (2)
AD × CF = CD × AE
⇒ 6 × 10 = D × 8
⇒ D = 60/8 = 7.5 cm
∴ AB = DC = 7.5 cm [∵ Opposite side of a parallelogram are equal]
Question: 3
Let ABCD be a parallelogram of area 124 cm2. If E and F are the mid-points of sides AB and CD respectively, then find the area of parallelogram AEFD.
Solution:
Given,
Area of a parallelogram ABCD = 124 cm2
Construction: Draw AP⊥DC
Proof:-
Area of a parallelogram AFED = DF × AP ⋅⋅⋅⋅⋅⋅⋅⋅ (1)
And area of parallelogram EBCF = FC × AP⋅⋅⋅⋅⋅⋅⋅⋅ (2)
And DF = FC ⋅⋅⋅⋅⋅ (3) [F is the midpoint of DC]
Compare equation (1), (2) and (3)
Area of parallelogram AEFD = Area of parallelogram EBCF
Question: 4
If ABCD is a parallelogram, then prove that
Ar (ΔABD) = Ar(ΔBCD) = Ar(ΔABC) = Ar(ΔACD) = (1/2) Ar (// gm ABCD).
Solution:
Given:-
ABCD is a parallelogram,
To prove: - Ar (ΔABD) = Ar(ΔBCD) = Ar(ΔABC) = Ar(ΔACD) = (1/2) Ar (//gm ABCD).
Proof:- We know that diagonal of a parallelogram divides it into two equilaterals .
Since, AC is the diagonal.
Then, Ar (ΔABC) = Ar(ΔACD) = (1/2) Ar(// gm ABCD) ⋅⋅⋅⋅ (1)
Since, BD is the diagonal.
Then, Ar(ΔABD) = Ar(ΔBCD) = (1/2) Ar(// gm ABCD) ⋅⋅⋅⋅⋅ (2)
Compare equation (1) and (2)
∴ Ar(ΔABC) = Ar(ΔACD) = Ar(ΔABD) = Ar(ΔBCD) = (1/2) Ar(// gm ABCD)..