Question

D bcm C
M
N
35. In a trapezium ABCD, AB|| DC, AB = a cm, and
DC = b cm. If M and N are the midpoints of the
nonparallel sides, AD and BC respectively then
find the ratio of ar(DCNM) and ar(MNBA).
B
A
a cm​

Answer 1

$\huge \fbox \green{Answer★}$

Join The diagonal DB Which Cuts the line MN at the point Y.

[look at the pic]

we know that M and N are the midpoints of AD and BC

$= MN||AB||CD$

$\text{ = Consider ∆ADB}$

• We know that M is the midpoints of AD and MY||AB

• So we get Y as the midpoint of BD it can be written as

$⇒MY = \frac{1}{2} AB$

$\text{⇒Consider∆ BDC}$

we know that

$⇒NY = \frac{1}{2} CD$

We Get

$⇒MN=MY+YN$

$\textbf{By Substituting the values}$

$⇒MN = \frac{1}{2} AB + \frac{1}{2} CD$

$\textbf{By taking out ½ as common}$

$⇒MN = \frac{1}{2} (AB + CD)$

• it is given that AB=a and CD=b so we get

$⇒MN = \frac{(a + b)}{2}$

• Construct DQ⊥AB where DQ cuts the line MN at the pòinť P.

So we know that P is the midpoint of DQ it can be written as

$⇒DP = PQ = h$

we get

$\text{Area of Trapezium} \: DCNM = \frac{1}{2} \times (MN + CD) \times DP$

$\textbf{By Substituting}$

$\text{Area of Trapezium} \: DCNM = \frac{1}{2}$

$= \left( \frac{a + b}{2} + a \right) \]h$

so we get

$\text{Area of Trapezium} \: DCNM = \frac{h}{4} (a + 3b)..(1)$

$\text{Area of Trapezium} \: MNBA = \frac{1}{2} \times (MN + AB) \times PQ$

$\textbf{By Substituting}$

$\text{Area of Trapezium} \: MNBA = \frac{1}{2} \left( \frac{a + b}{2} + a\right) \]h$

So we get

$\text{Area of Trapezium} \: MNBA = \frac{h}{4} (3a + b)..(2)$

$\textbf{By dividing both equations}$

$\text{Area of Trapezium} \: DCNM /$

$\text{Area of Trapezium} \: MNBA$

$= \frac{ \frac{h}{4}(a + 3b) }{ \frac{h}{4}(3a + b) }$

$\textbf{By Cancelling the similar term's}$

$\text{Area of Trapezium} \: DCNM /$

$\text{Area of Trapezium} \: MNBA$

$= \frac{(a + 3b)}{(3a + b)}$

it can be written as

Area of Trapezium DCNM:Area of Trapezium MNBA

$= (a + 3b):(3a + b)$

Therefore Ratio of area (DCNM) And area (MNBA) is

$(a + 3b):(3a + b)$

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Answer 2

Step-by-step explanation:

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