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Question
Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.
Solution 1
In order to do the desired construction,
we follow the following steps:
Step I: Take a point O on the plane of the paper and draw a circle of radius OA = 4 cm. Also, draw a concentric circle of radius OB = 6 cm.
Step II: Find the mid-point C of OB and draw a circle of radius OC = BC. Suppose this circle intersects the circle of radius 4 cm at P and Q.
Step III: Join BP and BQ to get the desired tangents from a point B on the circle of radius 6 cm.
By actual measurement, we find the
BP = BQ = 4.5 cm
Justification: In ∆BPO, we have
OB = 6 cm and OP = 4 cm
`OB^2 = BP^2 + OP^2 ` [Using Pythagoras theorem]
`\Rightarrow BP=\sqrt{OB^{2}-OP^{2}`
`\Rightarrow \sqrt{36-16}=\sqrt{20}=4.47 ~= 4.5 `
Similarly, BQ = 4.47cm ≈ 4.5 cm
Solution 2
Tangents on the given circle can be drawn as follows.
Step 1
Draw a circle of 4 cm radius with centre as O on the given plane.
Step 2
Draw a circle of 6 cm radius taking O as its centre. Locate a point P on this circle and join OP.
Step 3
Bisect OP. Let M be the mid-point of PO.
Step 4
Taking M as its centre and MO as its radius, draw a circle. Let it intersect the given circle at the points Q and R.
Step 5
Join PQ and PR. PQ and PR are the required tangents.
It can be observed that PQ and PR are of length 4.47 cm each.
In ΔPQO,
Since PQ is a tangent,
∠PQO = 90°
PO = 6 cm
QO = 4 cm
Applying Pythagoras theorem in ΔPQO, we obtain
PQ2 + QO2 = PQ2
PQ2 + (4)2 = (6)2
PQ2 + 16 = 36
PQ2 = 36 − 16
PQ2 = 20
PQ = `2sqrt5`
PQ = 4.47 cm
Justification
The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 4 cm). For this, let us join OQ and OR.
∠PQO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle.
∴ ∠PQO = 90°
⇒ OQ ⊥ PQ
Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle.
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