Quadrilaterals Test - 5

 Use pythagoras theorem in right triangle,

10² -[16/2]² = 100 -64 = 36 = [6]² ; hence the other diagonal = 6x2 = 12cm

The sum of the adjacent angles of a parallelogram is 180⁰.  Opposite sides of a parallelogram are equal. Hence it is a rectangle.

Let x be one angle. Then x+2/3 x=180 {sum of the adjacent angles of a parallelogram is 180⁰}

x=108, adjacent angle= 180-108= 72⁰

Area of Parallelogram = Base * Height

if 'a' is the side and 'b' is the base the height will be less than 'a' ( using Pythagorus theorem, a as Hypotenuse, h as height )

A< B

Join one of the diagonals which intersects the line joining mid points of the non parallel sides. This point of intersection become midpoint of the diagonal by using converse of midpoint theorem and we know that by mid point theorem that the line joining midpoints of any two sides of triangle measures half of the third side.so, by combining the results of two triangles we get above result.

Parallelograms on the same base and between the same parallels are equal in area. Hence the ration of two parallelograms will be 1:1.

I. The adjacent angles of a parallelogram are supplementary. Their halves add up to 90^o . so the angle bisectors enclose a right angle.

II. All the adjacent angle bisectors enclose right angles.Sowehave a rectangle being enclosed by the angle bisectors of a parallelogram.

III. The triangle formed by joining the mid-points of the sides of an isosceles triangle is always an isosceles triangle, because halves of equal sides are also equal

In square and rhombus, all sides are equal. By joining  the  points A and C diagonal AC formed. we get two traingles ABC and ADC which are congruent ( SAS conguence). Also, opposite sides are parallel. So, by using altenate angle property we can prove that angle BAC= angleDCA and angle DAC= angle ACB. But by CPCT angleDAC= angleBAC. So, all four angles made by diagonal AC with end points A and C  are equal which proves that diagonal AC bisects angle A and C both.

Given :

A B C is a triangle .

P Q R are the mid points of sides BC , CA nad AB .

AC = 21 cm

BC = 29 cm

AB = 30 cm

To find :

perimeter of quadrilateral ARQP ?

Q is the mid point of AC

P is the mid point of BC

QP is parallel to AB

QP = half of AB ( according to mid point theorem )

AB = 30 cm , QP = 15 CM ( QP is half of BA ) ( proved above )

R is the mid point of side AB

QP is also parallel to AR ( half of side AB )

PR is parallel to AC

PR = half of AC (according to mid point theorem )

AC = 21 cm , PR = 10.5 cm ( PR is half of AC ) ( proved above )

PR is parallel to AQ ( AQ is half of AC )

Since , in quadrilateral ARQP both the opposite sides are parallel it is a parallelogram.

Therefore , ARQP is a parallelogram .

WE know that

In parallelogram , opp sides are equal .

Therefore ,

PR = AQ = 10.5 cm

QP = AR = 15 cm

10.5 cm + 10.5 cm + 15 cm + 15 cm = 51 cm .

Therefore the perimeter of quadrilateral ARQP = 51 cm 

A quadrilateral with pair of opposite sides equal and having one right angle is called Rectangle and A rectangle with all sides equal is called Square. So, ABCD is a Square.