**Exercise 14.2**

QUESTION 1

**lf the given figure, ABCD is a parallelogram, AE ⊥ DC and CF ⊥ AD. If AB = 16 cm. AE = 8 cm and CF =10 cm, find AD.**

Sol :

**Given:** Here in the question it is given

(1) ABCD is a parallelogram,

(2) AE ⊥ DC and

(3) CF ⊥ AD , AB = 16 cm

(4) AE = 8 cm

**To find :** AD = ?

**Calculation :** We know that formula for calculating the

**Area of parallelogram = base × height**

Therefore,

Area of paralleogram ABCD = DC **×** AB (Taking base as DC and Height as AE)

Area of parallelogram ABCD = AB **× **AE (AB = DC as opposite side of the parallelogram are equal)

Therefore ,

Area of paralleogram ABCD =16 **×** 8 …(l)

Taking the base of Parallelogram ABCD as AD we get

Area of paralleogram ABCD = AD **×** CF (taking base as AD and height as CF)

Area of paralleogram ABCD = AD **×** 10 …(2)

Since equation (1) and (2) both represent the Area of the same Parallelogram ABCD , both should be equal.

Hence from equation (1) and (2)

This means that ,

16 **× **8 = AD **× **10

AD = 12.8 cm

Hence , we get the result as AD = 12.8 cm

QUESTION 2

**ln Q.No.1, if AD = 6 cm, CF = 10 cm, and AE = 8 cm, find AB.**

Sol :

**Given :** Here in the question it is given that

(1) ABCD is a parallelogram,

(2) AE ⊥ DC and

(3) CF ⊥ AD

(4) AD = 6 cm

(5) AB = 8 cm

(6) CF = 10 cm

**To find :** AB = ?

**Calculation : **We know that formula for calculating the

Area of paralleogram = base **×** height

Therefore,

Area of paralleogram ABCD = DC **×** AB (taking base as DC and Height as AE)

Area of parallelogram ABCD = AB **× **AE (AB = DC as opposite side o f the parallelogram are equal)

Therefore, Area of paralleograrm ABCD = 16 **×** 8

Area of Parallelogram ABCD = AB **×** 8 …(l)

Taking the base of Parallelogram ABCD as AD we get

Area of paralleogram ABCD = AD **×** CF (taking base as AD and height as CF)

Area of paralleogram ABCD = 6 **×** 10 …(2)

Since equation 1 and 2 both represent the Area of the same Parallelogram ABCD , both should be equal.

Hence equation (1) is equal to equation (2)

Which means that ,

AB **× **8 = 6 × 10

AD = 7.5 cm

Hence we got the measure of AB equal to 7.5 cm

QUESTION 3

**let ABCD be a paralllelogram of area 124 cm ^{2} . I f E and F are the mid-points of sides AB and CD respectively , then find the area of parallelogram AEFD .**

Sol :

**Given :** Here in the equation it is given that

(1) Area of paralleogram ABCD = 124 cm^{2}

(2) E is the mid-point of AB, which means

(3) F is the midpoint of CD, which means

**To find : **Area of parallleogram AEFD

**Calculation :** We know that formula for calculating the

**Area of parallelogram = base × height**

Therefore ,

Area of paralleogram ABCD = AB **×** AD (Taking base as AB and Height as AD) …(1)

Area of paralleogram ABFD = AE **×** AD (Taking base as AB and Height as AD) …(2)

= 62 cm^{2}

here we got the result Area of parallelogram AEFD = 62 cm^{2}

QUESTION 4

**If ABCD is a parallelogram, then prove that**

**area(ΔABD) = area(ΔBCD) = area(ΔABC) = area(ΔACD) **

Sol :

**Given :** Here in the question it is given that

(1) ABCD is parallelogram

To prove :

(1) Area of ΔADC

(2) Area of ΔBCD

(3) Area of ΔABC

(4) Area of ΔABD

**Construction :** Draw AE ⊥ CD

**Calculation :** We know that formula for calculating the

**Area of parallelogram = base × height**

Area of paralleogram ABCD = BC **×** AE (Taking base as BC and Height as AE ) …(l)

We know that formula for calculating the

Area of triangle

Area of ΔADC

(AD is the base of ΔADC and AE is the height of ΔADC)

[From equation-1]

Area of ΔADC

Similarly we can show that

(2) Area of ΔBCD

(3) Area of ΔABC

(4) Area of ΔABD