Question

Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is 60°. Also justify the construction. Measure the distance between the centre of the circle and the point of intersection of tangents.
A. 4 cm
B. 6 cm
C. 8 cm
D. 10 cm

Answer 1

Hint: For the purpose of drawing the tangents, we are going to follow the following steps, to get the answer. We will do construction and use the properties of triangle and circle in order to get the answer.

Complete step by step answer:
Steps of construction
1. First we will take a point O on the sheet and draw a circle of radius OA=4 cm.

2. Now make OA to B such that OA=AB=4 cm.

3. Taking A as the centre draws a circle of radius AO=AB=4 cm.

Now we suppose that it cuts the circle drawn in step 1 at P and Q.
4. Now we will join BP and BQ to get required tangents.

Mathematical steps:
As we can see
In ΔOAP
We have,
OA = OP = 4 cm (∴ this is radius of circle that we made with O as taking centre )
As we can see that AP = 4 cm (in Δ OAP)
∴ Δ OAP is Equilateral ...............................................................................................................(1)
⇒ ∠PAO = 60°
⇒ ∠BAP = 120°
In ΔBAP
Now we will get
BA=AP (By figure we can see ∠ABP = ∠APB = 30°) ................................................(2)
⇒ ∠PBQ = 60°
By using (2) , (1) we calculate BO i.e distance between centre of circle and point of intersection of tangents .....
BA = AP
And we also know that AO = AP =4 cm by equation (1)
⇒ BA = AO = 4cm ........................................................................................................................... (3)
 Now we can say that BO = BA + AO
 ⇒ BO = 4cm + 4cm
    Hence we will get answer BO = 8 cm (By using equation 3 )

Note: To solve these kinds of problem , First we have to make the rough diagram than , formulate the steps of construction then do the construction as step by step as we have formulated earlier , after construction , look for what is being asked to find , according to that use different properties , as we have used the property of equilateral in this question , After that solve the question by using mathematical calculations . Also we are careful in construction proper measurement should be taken in order to verify construction. After successful doing all the steps we will get the answer.