Question

pls ans ques 7..its from ch mid point theorem

Answer 1

Heya Friend !!

Here's ur answer wid solution :-

1 )

Given :-

ABCD is a square
EBC is an equilateral triangle

To find :- x

Solution :-

As we know that the side of square are equal.
and side of equilateral triangle are also equal.

Here Equilateral triangle is surmounted on square

So,

Side of square = side of equilateral ∆


DC = EC

hence ,

∆ECD is an isosceles triangle.

and ,

|_ CED = |_CDE


And all angle of square and equilateral triangle are 90° and 60° respectively.

So,

|_ ECD = |_BCE + |_BCD

|_ECD = 60° + 90°

|_ ECD = 150°


In ∆ECD using angle sum property of Triangle.

|_ECD + |_CED + |_CDE = 180

150 + 2 |_CED = 180

2|_CED = 180 - 150

2|_CED = 30

|_CED = 15


and

Now in ∆ BEC

|_BEC = 60°

|_BED + |_CED = 60°

x + 15 = 60

x = 60 - 15

x = 45


2)

Given :- |_ ECD = 146

As we know that diagonal of rectangle are equal and bisect each other.

AC = BD

AC/2 = BD/2

OC = OD

So, ∆ DOC is an isosceles triangle

|_ ODC = |_ OCD .. ( i )


|_ECD + |_ OCD = 180 ( by linear pair )

146 + |_COD = 180

|_OCD = 180 - 146

|_OCD = 34


|_ODC = 34 ... ( from i )


In ∆ DOC using exterior angle property

|_DOC + |_ODC = |_ECD

|_DOC = 146 - 34

|_DOC = 112

In ∆ AOB

|_AOB = |_DOC ( vertically opposite angle)

|_AOB = 112

Since opposite sides of rectangle are parallel

DC || AB

|_DCO = |_OAB ( alternate interior angle)

34 = |_OAB


|_CDO = |_OBA ( alternate interior angle )

34 = |_OBA



3 )

As we know that the diagonal of rhombus intersect at 90°

|_AOB = 90

|_DOA = 90

Let the angle be x
then

|_OAB = 3x
|_OBA = 2x

In ∆AOB using Angle sum property

|_OAB + |_OBA + |_AOB = 180

3x + 2x + 90 = 180

5x = 90

x = 18

|_OAB = 3x = 54
|_OBA = 2x = 36

As we know that diagonal of rhombus are angle bisector

|_ADO = |_ODC .... ( i )
|_DAO = |_OAB .....( ii )

From ii

|_DAO = 54

And opposite side of rhombus are parallel

So,

|_OBA = |_ CDO ( alternate interior angle)

36 = |_CDO

|_ADO = |_CDO ....( from i )

|_ADO = 36

In ∆AOD

|_ADO = 36
|_DOA = 90
|_DAO = 54

@Altaf