Question

In the given figure, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and
BC=4cm, then calculate the length of AP (in cm).​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

AP, AQ and BC are tangents to a circle touches the circle at P, Q and R respectively as shown in figure.

Further,

• AB = 5 cm

• AC = 6 cm

• BC = 4 cm

Now,

Consider,

Perimeter of triangle ABC,

$\rm :\longmapsto\:AB + BC + AC = 5 + 6 + 4$

$\rm :\longmapsto\:AB + BR + RC + AC = 15$

$\rm :\longmapsto\:AB + BP + CQ + AC = 15$

$\red{\bigg \{ \because \: Length \: of \: tangents \: are \: equal.\bigg \}}$

$\rm :\longmapsto\:AP + AQ = 15$

$\rm :\longmapsto\:AP + AP = 15$

$\red{\bigg \{ \because \: Length \: of \: tangents \: are \: equal.\bigg \}}$

$\rm :\longmapsto\:2AP= 15$

$\bf\implies \:AP = 7.5 \: cm$

Additional Information :-

1. Length of tangents drawn to a circle are equal from external point.

2. Radius and tangent are perpendicular to each other.

3. Tangents are equally inclined to the line segment joining the center and external point.

4. One and only one tangent can be drawn to a circle at a point.

5. A circle can have infinitely many tangents.