Question

Construct a tangent to a circle of radius 4 cm form a point on the concentric circle of radius 6 cm and measure its length. Also, verify the measurement by actual calculation.

Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm ?

Answer 1

Steps of Construction
Step 1: Mark a point O on the paper
Step 2: With O as center and radii 4cm and 6cm, draw two concentric circles.

Step 3: Mark a point P on the outer circle.
Step 4: Join OP.
Step 5: Draw the perpendicular bisector XY of OP, cutting OP at Q.
Step 6: Draw a circle with Q as center and radius OQ (or PQ), to intersect the inner circle in points T and T’.
Step 7: Join PT and PT’.

Here, PT and PT’ are the required tangents.
PT = PT’ 4.5 cm (Approx)
Verification by actual calculation
Join OT to form a right ΔOTP (Radius is perpendicular to the tangent at the point of contact)
In right ΔOTP,
` OP^2 =OT^2 + PT^2 `         (Pythagoras Theorem)

`⇒ PT = sqrt(OP^2 - OT^2)`

`⇒PT = sqrt(6^2 -4^2 )= sqrt(36-16) = sqrt(20)  ~~ 4.5 cm`

(OP = 6 cm and OT = 4cm)

Answer 2

Steps of construction:

1. Draw two concentric circles with centre O and radii 4 cm and 6 cm. Take a point P on the outer circle and then join OP.

2. Draw the perpendicular bisector of OP. Let the bisector intersects OP at M.

 

3. With M as the centre and OM as the radius, draw a circle. Let it intersect the inner circle at A and B.

 

4. Join PA and PB. Therefore, `\overline(PA)`  and `\overline(PB)`are the required tangents.