Help Plz!! Circle Geometry Question!?

Question

I got this question where you have to prove that a parralelogram that had all four of its vertices touching the circle was a rectangle.
I was wondering if anybody could explain how to do it.
Thanks

Answer 1

let parallelogram ABCD be inscribed inside a circle with centre O.
the adjacent angles of a parallelogram are supplemetary
so, LA+LB=180.....(1)
LB+LC=180......(2)
Since each of the four point s A,B,C,D fall on the circle, so the parallelogram is a cyclic quadrilateral
as per the properties of a cyclic quad. the opp. angles are supplemetary
so, LA+LC=180....(3)
LB+LD=180.......(4)
calculate (2)-(3)
LB-LA=180
so, LB=LA....(5)
put (5) in (1)
LA+LA=180
2LA=180
LA=90
from (5) we can say
LA=LB=90
putting the value of LA & LLB in (3) & (4) we get LC=LD=90
so all fou r angles of the paralleloogram are 90 degrees hence it is a rectangle

Answer 2

For any parallelogram adjacent angles are supplementary. This is because each side of a parallelogram is a transverse line intersecting two parallel lines. Interior angles on the same side are supplementary.

In an inscribed parallelogram opposite angles together subtend the entire circle. Therefore they are also supplmentary.

Put together adjacent angles are x and 180 - x.
Opposite angles are also x and 180 - x.

Therefore

180 - x = x
180 = 2x
x = 90°

So an inscribed parallelogram must necessarily be a rectangle.

Answer 3

Let ABCD be a concyclic parallelogram.

In any paralleogram, adjacent angles add up to 180 degrees.

Hence, A + B = 180 degrees ... ... ( 1 )

In a concyclic paralleogram, opposite angles also add up to 180 degrees.

Hence, A + C = 180 degrees ... ... ( 2 )

From ( 1 ) and ( 2 ), B = C.

But, B and C being adjacent angles, B + C = 180 degrees.

Therefore, B + B = 180 degrees => B = 90 degrees.

Similarly, A, C and D are 90 degrees.

Hence ABCD is a rectangle.

Answer 4

First off, lets recall what we know about parallelograms. From wikipedia:

"parallelogram is a quadrilateral with two sets of parallel sides. The opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent."

This means opposite angles are equal.

Then, lets consider any quadrilateral that has its vertices on a circle. Well known circle theorem states that:

" If a quadrilateral is inscribed in a circle, then both pairs of opposite angles are supplementary."

Which means opposite angles must add up to 180 degrees.

It is clear, combining these two theorems, we can conclude that the two angles opposite each other, which must be equal, must add up to give 180 degrees, or

x + x = 180
=> 2x = 180
=> x = 90 degrees

Performing this logic on both pairs of opposite angles proves that all angles of the parallelogram must be 90 degrees, and therefore it is a rectangle. As required.

Answer 5

first we need to prove that the angles of the paralellogram are equal to 90 degrees by joining the diagonals first. then prove the congruence of the triangles formed [satisfies sss congruence]. thus we also get that the angles are also equal to each other.then add all the angles and equate it to 360. since all the angles are = , they result in each angle to be 90. thus it is a rectangle

Answer 6

see in a parallelogram adj.sides r equal...and opp.sides r equal in a quadrilateral when its four vertices touch the circle(cyclic)...therfore all the angles will be 90...therefore it is a rectangle...

Answer 7

i really dont know, try bbc bitesize