Two circles intersect at two points $A$ and $B$. $AD$ and $AC$ are diameters to the two circles (see Fig.). Prove that $B$ lies on the line segment $DC$.
(N/A) Join $AB.$
$\angle ABD = 90^\circ$ (Angle in a semicircle is a right angle).
$\angle ABC = 90^\circ$ (Angle in a semicircle is a right angle).
Adding these two equations,we get:
$\angle ABD + \angle ABC = 90^\circ + 90^\circ = 180^\circ.$
Since the sum of the angles is $180^\circ$,the points $D, B,$ and $C$ form a straight line. Therefore,$B$ lies on the line segment $DC.$
$\angle ABD = 90^\circ$ (Angle in a semicircle is a right angle).
$\angle ABC = 90^\circ$ (Angle in a semicircle is a right angle).
Adding these two equations,we get:
$\angle ABD + \angle ABC = 90^\circ + 90^\circ = 180^\circ.$
Since the sum of the angles is $180^\circ$,the points $D, B,$ and $C$ form a straight line. Therefore,$B$ lies on the line segment $DC.$
Explore More
Similar Questions
In the figure,$A, B$ and $C$ are three points on a circle with centre $O$ such that $\angle BOC = 30^{\circ}$ and $\angle AOB = 60^{\circ}$. If $D$ is a point on the circle other than the arc $ABC$,find $\angle ADC$. (in $^{\circ}$)
Medium
View SolutionIf two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection,prove that the chords are equal.
Medium
View SolutionDraw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?
Medium
View Solution$AB$ is a chord of a circle with centre $P$. Point $C$ is a point other than $A$ and $B$ on the major arc $AB$. If $\angle ACB + \angle APB = 135^{\circ}$,then find $\angle ACB$ and $\angle APB$.
Difficult
View SolutionTwo circles of radii $5\, cm$ and $3\, cm$ intersect at two points and the distance between their centres is $4\, cm.$ Find the length of the common chord. (in $, cm$)
Difficult
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo