If a hexagon ABCDEF circumscribes a circle,pr | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(N/A) Let the points of contact of the circle with the sides $AB, BC, CD, DE, EF,$ and $FA$ be $P, Q, R, S, T,$ and $U$ respectively.Since the lengths

Vedclass · Accepted Answer

Vedclass · Answer

(N/A) Let the points of contact of the circle with the sides $AB, BC, CD, DE, EF,$ and $FA$ be $P, Q, R, S, T,$ and $U$ respectively.
Since the lengths of tangents drawn from an external point to a circle are equal,we have:
$AP = AU$
$BP = BQ$
$CQ = CR$
$DR = DS$
$ES = ET$
$FT = FU$
Now,consider the sum of alternate sides:
$AB + CD + EF = (AP + PB) + (CR + RD) + (ET + TF)$
Substituting the tangent equalities:
$AB + CD + EF = (AU + BQ) + (CQ + DS) + (ES + FU)$
Rearranging the terms:
$AB + CD + EF = (BQ + CQ) + (DS + ES) + (FU + AU)$
$AB + CD + EF = BC + DE + FA$
Hence,it is proved.

If a hexagon $ABCDEF$ circumscribes a circle,prove that $AB + CD + EF = BC + DE + FA$.

Similar Questions

The incircle of $\Delta ABC$ touches the sides $\overline{AB}$, $\overline{BC}$ and $\overline{CA}$ at points $P$, $Q$ and $R$ respectively. If $AB = 14$, $BC = 11$ and $CA = 7$, find the lengths of $AP$, $BQ$ and $RC$.

If $d_{1}$ and $d_{2}$ $(d_{2} > d_{1})$ are the diameters of two concentric circles and $c$ is the length of a chord of the outer circle which is tangent to the inner circle,prove that $d_{2}^{2} = c^{2} + d_{1}^{2}$.

Write 'True' or 'False' and give reasons for your answer.
The angle between two tangents to a circle may be $0^{\circ}$.

$A$ line that intersects a circle at two distinct points is called a $\ldots \ldots \ldots \ldots$ of the circle.

$P$ is in the exterior of $\odot(O, 30)$. The tangent drawn from $P$ to the circle touches the circle at $Q$. If $OP = 34$,then $PQ = \dots$

Vedclass Test Series

Exam Paper Generator

Online Exam Module