Home
Class 9
MATHS
Prove that any four vertices of a regula...

Prove that any four vertices of a regular pentagon are concyclic.

Text Solution

Verified by Experts

Given : A regular pentagon `ABCDE`.
To Prove : Every set of four vertices of `ABCDE` is a set of points lying on a circle.
Proof : First we show that the points `A,B,C` and `E`, lies on a circle.
Join `AC` and `BE`.
In `Delta ABC` and `Delta BAE`, we have

`AB` `=` `BA` [common ]
`BC` `=` `AE` [sides of a regular pentagon]
`/_ ABC = /_ BAE` [ each equal to `108^(@) ]`
`:. Delta ABC ~= Delta BAE ` [ by SAS -congruence]
`implies /_ BCA = /_ AEB ` [ c.p.c.t.]
Thus, `AB` subtends equal angles at two points `C` and `E` on the same side of `AB`.
`:. ` the points `A,B,C,E` are concyclic. [ by Theorem 5 at Page 444].
Similary , every set of four vertices of pentagon ABCDE is a set of concyclic points.
Promotional Banner

Similar Questions

Explore conceptually related problems

Symmetry of a regular pentagon.

Lines of symmetry of a regular pentagon

Four particles each having a charge q are placed on the four vertices of a regular pentagon. The distance of each comer from !be centre is 'a'. Find the electric field at the centre of pentagon.

Five point masses m each are kept at five vertices of a regular pentagon. Distance of centre of pentagon from any one of the verticle is 'a'. Find gravitational potential and filed strength at centre.

Find each interior of a regular pentagon.

Five identical charges, q_1 each, are placed at the vertices of a regular pentagon having side length l_1 . The net electrostatic force on any of the charges due to other four is F_1 . Find the electrostatic force F_2 on any one of the five identical charges, q_2 each, placed at the vertices of a regular pentagon having side length l_2 .