# Class 10 RD Sharma Solution – Chapter 11 Constructions – Exercise 11.3

**Question 1. Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.**

**Solution:**

Follow these steps for construction :

Step 1:Construct of 6 cm radius with centre O.

Step 2:Mark a point P, 10 cm away from the centre O.

Step 3:Now join PO then bisect it at M.

Step 4:From the centre M and the diameter PO, construct a circle that will be intersecting the given circle at T and S.

Step 5:Thus join PT and PS.Further PT and PS are the required tangents.

**Question 2. Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.**

**Solution:**

Follow these steps for construction :

Step 1:Construct of radius 3 cm with the centre O.

Step 2:Construct a diameter.

Step 3:Mark two points P and Q on this diameter with a distance of 7 cm each from the centre O, as shown below.

Step 4:Now bisect QO at N and PO at M.

Step 5:From the centres M and N, construct circle on ‘PO’ and ‘QO’ as diameter which intersect the given circle at S, T and S’, T’ respectively.

Step 6:Further join PS, PT, QS’ and QT’.Therefore, PS, PT, QS’ and QT’ are the required tangents to the given circle.

**Question 3. Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle. [CBSE 2013]**

**Solution:**

Follow these steps for construction:

Step 1:Construct a line segment AB of 8 cm.

Step 2:Draw circles from the centre A and radius 4 cm and with centre B and radius 3 cm.

Step 3:Now bisect AB at M.

Step 4:From centre M and diameter AB, construct a circle which will intersects the two circles at S’, T’ and S, T respectively.

Step 5:Further join AS, AT, BS’and BT’.Therefore, AS, AT, BS’ and BT’ are the required tangent.

**Question 4. Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of 6.2 cm from its centre.**

**Solution:**

Follow these steps for construction :

Step 1:Construct a circle of radius 3.5 cm with centre O.

Step 2:Mark a point P which will be of 6.2 cm from O.

Step 3:Now bisect PO at M and construct a circle with centre M and diameter OP which will intersects the given circle at T and S respectively.

Step 4:Further join PT and PS.Therefore, PT and PS are the required tangents to circle.

**Question 5. Draw a pair of tangents to a circle of radius 4.5 cm, which are inclined to each other at an angle of 45°. [CBSE 2013]**

**Solution:**

Follow these steps for construction:

At centre the angle is 180° – 45° = 135°

Step 1:Construct a circle of radius 4.5 cm with centre O.

Step 2;Now, at O, construct an angle ∠TOS = 135°

Step 3:Further at T and S draw perpendicular which will meet at P.Therefore, PT and PS are the tangents which inclined each other 45°.

**Question 6. Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle.**

**Solution:**

Follow these steps for construction:

Step 1:Construct a line segment BC of 8 cm

Step 2:From B construct an angle of 90°

Step 3:Construct an arc BA˘ of 6cm cutting the angle at A.

Step 4:Now join AC. Thus, ΔABC is the required A.

Step 5:Construct perpendicular bisector of BC cutting BC at M.

Step 6:Mark M as centre and BM as radius, construct a circle.

Step 7:Mark A as centre and AB as radius, construct an arc cutting the circle at E. Thus, join AE.Therefore, AB and AE are the required tangents.

Justification:

Given: ∠ABC = 90°

Thus, OB is a radius of the circle.

Therefore, AB is a tangent to the circle.

Also AE is a tangent to the circle.

**Question 7. Draw two concentric circles of radii 3 cm and 5 cm. Construct a tangent to the smaller circle from a point on the larger circle. Also, measure its length. [CBSE 2016]**

**Solution:**

Given: Two concentric circles of radii 3 cm and 5 cm with centre O.

We have to construct a pair of tangents from point P on outer circle to the other.

Follow these steps for construction:

Step 1:Construct two concentric circles of radii 3 cm and 5 cm with the centre O.

Step 2:Then, take any point P on outer circle and join OP.

Step 3:Now, bisect OP and let M’ be the mid-point of OP.Take M’ as centre and OM’ as radius construct a circle dotted which will cut the inner circle as M and P’.

Step 4:Further, join PM and PP’. Therefore, PM and PP’ are the required tangents.

Step 5:After measuring PM and PP’, we find that PM = PP’ = 4 cm.Actual calculation:

In right angle ΔOMP, ∠PMO = 90°

Therefore,

PM

^{2}= OP^{2}– OM^{2 }{by Pythagoras theorem i.e. (hypotenuse)^{2}= (base)^{2}+ (perpendicular)^{2}}⇒ PM

^{2}= (5)^{2}– (3)^{2}= 25 – 9 = 16⇒ PM = 4 cm

Hence, the length of both tangents is 4 cm.

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