Question

In the given figure, AP and BP are tangents to a circle with center O, such that AP = 5cm and $\angle APB={{60}^{\circ }}$. Find the length of the chord AB.

Answer 1

Hint: Here, we are given a diagram in which AP and BP are two tangents with AP = 5cm and $\angle APB={{60}^{\circ }}$. We need to find the length of the chord AB. For this, we will use following properties:
(i) The length of the tangents drawn from an external point to a circle are always equal.
(ii) Isosceles triangle property: the angles corresponding to equal sides of the triangle are also equal.
(iii) If all the angles of a triangle are equal (${{60}^{\circ }}$ each) the triangle is an equilateral triangle.

Complete step by step answer:
Let us redraw the diagram. Here AP = 5cm, $\angle APB={{60}^{\circ }}$ and we need to find the length of the chord AB.

As we can see, AP and BP are tangents to the given circle from the same external point P. So we can use the property that, length of the tangents drawn from an external point to a circle are equal. Hence for this figure we get PA = PB.
Now we can see that APB forms a triangle with equal sides PA and PB.
We know that, in an isosceles triangle, angles corresponding to equal sides are also equal. Therefore, $\angle PAB=\angle PBA$.
We are given $\angle APB={{60}^{\circ }}$.
Using angle sum property of the triangle $\angle APB+\angle PAB+\angle PBA={{180}^{\circ }}$.
Let us suppose that, $\angle PAB=\angle PBA=x$ so we get ${{60}^{\circ }}+x+x={{180}^{\circ }}\Rightarrow {{60}^{\circ }}+2x={{180}^{\circ }}\Rightarrow 2x={{120}^{\circ }}$.
Dividing by 2 both sides we get: $x={{60}^{\circ }}$.
Since x was supposed to be equal to $\angle PAB=\angle PBA={{60}^{\circ }}$.
Since all the angles of the triangle are equal to ${{60}^{\circ }}$. Therefore, the triangle is an equilateral triangle.
$\Delta APB$ is an equilateral triangle. So, AP = PB = AB.
We are given AP as 5cm.
Therefore, AP = PB = AB = 5cm.

Hence, AB = 5cm which is the required length of the chord.

Note: Students should keep in mind all the properties before doing these sums. We can only draw two tangents from an external point to a circle and these tangents are always of equal length. An equilateral triangle always has angles of ${{60}^{\circ }}$ each.