Question

In the fig find the length of the chord AB if PB=6cm and anglePBA - 60
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Answer 1

Answer:

Here, we are given a diagram in which AP and BP are two tangents with AP = 5cm and ∠APB=60∘. 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∘ each) the triangle is an equilateral triangle.

Complete step by step answer:

Let us redraw the diagram. Here AP = 5cm, ∠APB=60∘ 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, ∠PAB=∠PBA.

We are given ∠APB=60∘.

Using angle sum property of the triangle ∠APB+∠PAB+∠PBA=180∘.

Let us suppose that, ∠PAB=∠PBA=x so we get 60∘+x+x=180∘⇒60∘+2x=180∘⇒2x=120∘.

Dividing by 2 both sides we get: x=60∘.

Since x was supposed to be equal to ∠PAB=∠PBA=60∘.

Since all the angles of the triangle are equal to 60∘. Therefore, the triangle is an equilateral triangle.

ΔAPB 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.