**Exercise 11.1**

**1) Draw a line segment of length 7.6 cm and divide it in 5: 8 ration. In addition, find the measure of two parts.**

**Solution:**

Procedure for construction:

- Draw any ray AX, making an acute angle with AB.
- Locate 13(= 5 + 8) points A
_{1}, A_{2}, A_{3}……… A13 on AX so that AA_{1}= A_{1}A_{2}A_{l2}A_{13}. - Join BA
_{13}. - Through the point A5(m = 5), draw a line parallel to BA13 (by making an angle equal to L AA13 B at A5 intersecting AB at C. Then AC: CB = 5 : 8)

**2) Construct a triangle with sides 4 cm, 5 cm and 6 cm and then a similar triangle to it whose sides are 2/3 of the corresponding sides of the first one.**

**Solution:**

Procedure for construction:

- Draw a line segment BC with length 5 cm.
- With B as centre and radius of 4 cm draw an arc.
- With C as centre and radius of 6 cm draw an arc.
- Join AB and AC. Then, ∆ABC is the required triangle.
- Below BC, make an acute angle ∠CBX
- Along BX, mark up three points B1, B2, B3 such that BB1 = B
_{1}B_{2}= B_{2}B_{3} - Join B
_{3}C - From B2, draw B
_{2}C’llB_{3}c, meeting BC at C’ - From C’ draw C’ All CA, meeting BA at A’
- Then ∆A’BC’ is the required triangle, each of whose sides is two-third of the corresponding sides of ∆ABC.

**3) Construct a triangle with side lengths 5 cm, 6 cm and 7 cm and then another triangle whose sides are of the corresponding sides of the first triangle.**

**Solution:**

Procedure for construction :

- Draw a line segment BC with length 6 cm.
- With B as centre and keeping radius as 5 cm, draw an arc.
- With C as centre and keeping radius as 7 cm, draw another arc, intersecting the previously drawn arc at Point A.
- Join AB and AC. Then, ∆ABC is the required triangle.
- Below BC, make an acute angle∠CBX.
- Along BX, mark up seven points B
_{1}, B_{2}, B_{3}….. B_{7}such that BB_{1}= B_{1},B_{2}, B_{6}B_{7}. - Join B
_{5}to C (5 being smaller of 5 and 7 in7/5) and draw a line through B_{7}parallel to B_{5}C, intersecting the extended line segment BC at C’. - Draw a line through C’ parallel to CA intersecting the extended line segment BA at A’. Then A’BC’ is the required triangle.

**4) Draw a triangle ABC with sides BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are ¾ of the corresponding sides of ∆ABC.**

**Solution:**

**Procedure for construction:**

(i) Draw a triangle ABC with BC = 6 cm, AB = 5 cm and ∠ABC = 60°.

(ii) Draw any ray BX making an acute angle with BC on the side opposite to the vertex X.

(iii) Locate 4(the greater of 3 and 4 in ¾) points B1, B2, B3, B4 on BX so that BB_{1} = B_{1}B_{2} = B_{2}B_{3} = B_{3}B_{4}.

(iv) Join B_{4}C and draw a line through B3(the 3rd point, 3 being smaller of 3 and 4 in ¾) parallel to B_{4}C to intersect BC at C’.

(v) Draw a line through C’ parallel to the line CA to intersect BA at A’. Then ∆A’BC’ is the required triangle.

Justification of construction

∆ABC ~ ∆A’BC’ , Therefore,

\(\frac{AB}{A’B} = \frac{AC}{A’C’} = \frac{BC}{BC’}\)But, \(\frac{BC}{BC’} = \frac{BB_{3}}{BB_{4}} = \frac{3}{4}\)

So, \(\frac{AB}{A’B} = \frac{AC}{A’C’} = \frac{BC}{BC’} = \frac{3}{4}\)

**5) Draw a triangle ABC with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then, construct a triangle whose sides are4/3 times the corresponding sides of ∆ABC.**

**Solution:**

**Procedure for construction : **

(i) Draw a triangle ABC with BC = 7cm, ∠B = 45° and ∠A = 105°.

(ii) Draw any ray BX making an acute angle with BC on the side opposite to the vertex X.

(iii) Locate 4(the greater of 3 and 4 in 4/3) points B_{1}, B_{2}, B_{3}, B_{4} on BX so that 3 BB_{1} = B_{1} B_{2} = B_{2}B_{3} = B_{3}B_{4}.

(iv) Join NC’ and draw a line through B3(the 3rd point, 3 being smaller of 3 and 4 in 1) parallel to NC’ to intersect BC’ at C. 3

(v) Draw a line through C’ parallel to the line CA to intersect BA at A’. Then A NBC’ is the required triangle.

**6) Construct a triangle of isosceles type, whose base is 8 cm and height 4 cm and then another triangle whose sides are 1.5 times the corresponding sides of the isosceles triangle.**

**Solution:**

**Given**:An isosceles triangle whose base is 8 cm and height 4 cm. Scale factor: 1 =

**Required:** To construct a similar triangle to above whose sides are 1.5 times the above triangle.

**Procedure for construction:**

(i) Draw a line segment BC = 8 cm.

(ii) Draw a perpendicular bisector AD of BC.

(iii) Join AB and AC we get a isosceles ∆ABC.

(iv) Construct an acute angle∠CBX downwards.

(v) On BX make 3 equal parts.

(vi) Join C to B_{2} and draw a line through B_{3} parallel to B_{2}C intersecting the extended line segment BC at C’.

(vii) Again draw a parallel line C’A’ to AC cutting BP at A’.

(viii) ∆A’BC’ is the required triangle.

**Exercise 11.2**

**7) Draw a circle with 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:**

**Procedure for construction:**

- Draw a line segment of length AB = 10 cm. Bisect AB by constructing a perpendicular bisector of AB. Let M be the mid-point of AB.
- With M as centre and AM as radius, draw a circle. Let it intersect the given circle at the points P and Q.
- Join PB and QB. Thus, PB and QB are the required two tangents.

**Justification:** Join AP. Here ∠APB is an angle in the semi-circle. Therefore, ∠APB = 90°. Since AP is a radius of a circle, PB has to be a tangent to a circle. Similarly, QB is also a tangent to a circle.

In a Right ∆APB, AB^{2} = AP^{2} + PB^{2} (By using Pythagoras Theorem)

PB^{2} = AB^{2} – AP^{2} = 10^{2} — 6^{2} = 100 – 36 = 64

PB = 8 cm.

**8) 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:**

**Procedure for construction:**

- Draw a line segment of length OA = 4 cm. With O as centre and OA as radius, draw a circle.
- With O as centre draw a concentric circle of radius 6 cm(0B).
- Let C be any point on the circle of radius 6 cm, join OC.
- Bisect OC such that M is the mid point of OC.
- With M as centre and OM as radius, draw a circle. Let it intersect the given circle of radius 4 cm at the points P and Q.
- Join CP and CQ. Thus, CP and CQ are the required two tangents.

**Justification:**

Join OP. Here ∠OPC is an angle in the semi-circle. Therefore, ∠OPC = 90°. Since OP is a radius of a circle, CP has to be a tangent to a circle. Similarly, CQ is also a tangent to a circle.

In ∆COP, ∠P = 90°

\(CO^{2} = CP^{2} + OP^{2}\) \(CP^{2} = CO^{2} – OP^{2}\)=\(6^{2} – 4^{2}\)

\(CP = 2\sqrt{5}cm\)

**9) Draw a circle with radius 3 cm. On one of its extended diameter, take two points P and Q each at a distance of 7 cm from its centre. From two points P and Q, draw tangents to the circle.**

**Solution:**

**Given:**

Two points P and Q on the diameter of a circle with radius 3 cm OP = OQ = 7 cm.

**Aim**:

To construct the tangents to the circle from the given points P and Q.

**Procedure for construction:**

- Draw a circle with radius 3 cm with centreO.
- Extend its diameter both the sides and cut OP = OQ = 7 cm.
- Bisect OP and OQ.Let mid-points of OP and OQ be M and N.
- With M as centre and OM as radius, draw a circle. Let it intersect (0, 3) at two points A and B. Again taking N as centre ON as radius draw a circle to intersect circle(0, 3) at points C and D.
- Join PA, PB, QC and QD. These are the required tangents from P and Q to circle (0, 3).

**10) Draw a pair of tangents to a circle which is of radius 5 cm, such that they are inclined to each other at an angle of 60°.**

**Solution:**

**To determine**: To draw tangents at the ends of two radius which are inclined to each other at 120°

**Procedure for construction : **

- Keeping O as centre, draw a circle of radius 5 cm.
- Take a point Q on the circle and join it to O.
- From OQ, Draw∠QOR = 120°.
- Take an external point P.
- Join PR and PQ perpendicular to OR and OQ respectively intersecting at P.

The required tangents are RP and QP.

it’s helpful

No justification

Best

Its very helpful

Nice explanation…

Answers for the last two questions of 11.2 are not there

Helpful to me