Tangents and Intersecting Chords

### ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6cm (not drawn to scale). Three circles are drawn touching each other with the vertices as their centers. Find the radii of the three circles.

Solution: According to the given question, ABC is a triangle with $AB\text{ }=\text{ }10\text{ }cm,\text{ }BC=\text{ }8\text{ }cm,\text{ }AC\text{ }=\text{ }6\text{ }cm$ Three circles are drawn...

### In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If angle ACO = 30degree, find: angle APB

Solution: In the given fig, $O$is the centre of the circle and, $CA\text{ }and\text{ }CB$are the tangents to the circle from $C$Also, $\angle ACO\text{ }=\text{ }{{30}^{o}}$ $P$ is any...

### In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If angle ACO = 30degree, find: (i) angle BCO (ii) angle AOB

Solution: In the given fig, $O$is the centre of the circle and, $CA\text{ }and\text{ }CB$are the tangents to the circle from $C$Also, $\angle ACO\text{ }=\text{ }{{30}^{o}}$ $P$ is any...

### In the figure, AB is the chord of a circle with centre O and DOC is a line segment such that BC = DO. If ∠C = 20degree, find angle AOD.

Solution: Join $OB$ In $\vartriangle OBC$ we have $BC\text{ }=\text{ }OD\text{ }=\text{ }OB$[Radii of the same circle] $\angle BOC\text{ }=\angle BCO\text{ }=\text{ }{{20}^{o}}$ And ext....

### Bisectors of vertex angles A, B and C of a triangle ABC intersect its circumcircle at points D, E and F respectively. Prove that angle EDF = 90degree – ½ ∠A

Join $ED,\text{ }EF\text{ }and\text{ }DF$ And $BF,\text{ }FA,\text{ }AE\text{ }and\text{ }EC$ $\angle EBF\text{ }=\angle ECF\text{ }=\angle EDF\text{ }\ldots ..\text{ }\left( i \right)$ [Angle...

### Show that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base.

Join $AD$ And $AB$ is the diameter. We have $\angle ADB\text{ }=\text{ }90{}^\text{o}~$[Angle in a semi-circle] But,  $\angle ADB\text{ }+~\angle ADC\text{ }=\text{ }180{}^\text{o}~$[Linear...

### Prove that, of any two chords of a circle, the greater chord is nearer to the center.

Given: $A$ circle with center $O$ and radius $r$ $AB\text{ }and\text{ }CD$are two chords such that $AB\text{ }>\text{ }CD$ Also, $OM\bot AB\text{ }and\text{ }ON\bot CD$ Required to...

### Tangent at P to the circumcircle of triangle PQR is drawn. If this tangent is parallel to side QR, show that triangle PQR is isosceles.

Let $DE$be the tangent to the circle at $P$ And, $DE\text{ }||\text{ }QR$[Given] $\angle EPR\text{ }=\angle PRQ$[Alternate angles are equal] $\angle DPQ\text{ }=\angle PQR$[Alternate...

### If the sides of a parallelogram touch a circle, prove that the parallelogram is a rhombus.

Solution: Suppose a circle touch the sides $AB,\text{ }BC,\text{ }CD\text{ }and\text{ }DA$of parallelogram $ABCD\text{ }at\text{ }P,\text{ }Q,\text{ }R\text{ }and\text{ }S$respectively. Now,...

### If the sides of a quadrilateral ABCD touch a circle, prove that AB + CD = BC + AD.

Solution: Suppose, a circle touch the sides $AB,\text{ }BC,\text{ }CD\text{ }and\text{ }DA$ of quadrilateral $ABCD\text{ }at\text{ }P,\text{ }Q,\text{ }R\text{ }and\text{ }S$respectively. As,...