NCERT Options for Class 10 Maths Chapter 11 Constructions

Constructions: Train 11.1

In every of the next, give the justification of the development additionally:

Query 1. Draw a line phase of size 7.6 cm and divide it within the ratio 5: 8. Measure the 2 elements.

Resolution:

Steps of development:

To divide the road phase of seven.6 cm within the ratio of 5 : 8.

Step 1. Draw a line phase AB of size 7.6 cm.

Step 2. Draw a ray AC which types an acute angle with the road phase AB.

Step 3. Mark the factors = 13 as (5+8=13) factors, comparable to A₁, A₂, A₃, A₄ …….. A₁₃, on the ray AC such that it turns into AA₁ = A₁A₂ = A₂A₃ and such like this.

Step 4. Now be a part of the road phase and the ray, BA₁₃.

Step 5. Therefore, the purpose A₅, assemble a line parallel to BA₁₃ which makes an angle equal to ∠AA₁₃B.

Step 6. Level A₅ intersects the road AB at level X.

Step 7. X is that time which divides line phase AB into the ratio of 5:8.

Step 8. Thus, measure the lengths of the road AX and XB. Therefore, it measures 2.9 cm and 4.7 cm respectively.

Justification:

The development will be justified by proving that

frac{AX}{XB} = frac{5}{ 8}

From development, we now have A₅X || A₁₃B. By the Primary proportionality theorem for the triangle AA₁₃B, we are going to get

frac{AX}{XB} =frac{AA_5}{A_5A_{13}}         ….. (1)

By the determine we now have constructed, it may be seen that AA₅ and A₅A₁₃ comprises 5 and eight equal divisions of line segments respectively.

Thus,

frac{AA_5}{A_5A_{13}}=frac{5}{8}         … (2)

Evaluating the equations (1) and (2), we get

frac{AX}{XB} = frac{5}{ 8}

Thus, Justified.

Query 2. Assemble a triangle of sides 4 cm, 5 cm and 6 cm after which a triangle just like it whose sides are 2/3 of the corresponding sides of the primary triangle.

Resolution:

Steps of Building:

Step 1. Draw a line phase XY which measures 4 cm, So XY = 4 cm.

Step 2. Taking level X as centre, and assemble an arc of radius 5 cm.

Step 3. Equally, from the purpose Y as centre, and draw an arc of radius 6 cm.

Step 4. Thus, the next arcs drawn will intersect one another at level Z.

Step 5. Now, we now have XZ = 5 cm and YZ = 6 cm and subsequently ΔXYZ is the required triangle.

Step 6. Draw a ray XA which is able to make an acute angle alongside the road phase XY on the other aspect of vertex Z.

Step 7. Mark the three factors comparable to X₁, X₂, X₃ (as 3 is larger between 2 and three) on line XA such that it turns into XX₁ = X₁X₂ = X₂X₃.

Step 8. Be part of the purpose YX₃ and assemble a line via X₂ which is parallel to the road YX₃ that intersect XY at level Y’.

Step 9. From the purpose Y’, assemble a line parallel to the road YZ that intersect the road XZ at Z’.

Step 10. Therefore, ΔXY’Z’ is the required triangle.

Justification:

The development will be justified by proving that

XY’ = frac{2}{3}XY Y’Z’ = frac{2}{3}YZ XZ’= frac{2}{3}XZ

From the development, we get Y’Z’ || YZ

∴ ∠XY’Z’ = ∠XYZ (Corresponding angles)

In ΔXY’Z’ and ΔXYZ,

∠XYZ = ∠XY’Z (Proved above)

∠YXZ = ∠Y’XZ’ (Frequent)

∴ ΔXY’Z’ ∼ ΔXYZ (From AA similarity criterion)

Subsequently, 

frac{XY’}{XY} = frac{Y’Z’}{YZ}= frac{XZ’}{XZ}          …. (1)

In ΔXXY’ and ΔXXY,

∠X₂XY’ =∠X₃XY (Frequent)

From the corresponding angles, we get,

∠AA₂B’ =∠AA₃B

Thus, by the AA similarity criterion, we get

ΔXX₂Y’ and XX₃Y

So, frac{XY’}{XY} = frac{XX_2}{XX_3}

Subsequently, frac{XY’}{XY} = frac{2}{3}          ……. (2)

From the equations (1) and (2), we acquire

frac{XY’}{XY}=frac{Y’Z’}{YZ} = frac{XZ’}{ XZ} = frac{2}{3}

It’s written as

XY’ = frac{2}{3}XY

Y’Z’ = frac{2}{3}YZ

XZ’=  frac{2}{3}XZ

Subsequently, justified.

Query 3. Assemble a triangle with sides 5 cm, 6 cm, and seven cm after which one other triangle whose sides are 7/5 of the corresponding sides of the primary triangle

Resolution:

Steps of development:

Step 1. Assemble a line phase XY =5 cm.

Step 2. By taking X and Y as centre, and assemble the arcs of radius 6 cm and 5 cm respectively.

Step 3. These two arcs will intersect one another at level Z and therefore ΔXYZ is the required triangle with the size of sides as 5 cm, 6 cm, and seven cm respectively.

Step 4. Assemble a ray XA which is able to make an acute angle with the road phase XY on the other aspect of vertex Z.

Step 5. Pinpoint the 7 factors comparable to X₁, X₂, X₃, X₄, X₅, X₆, X₇ (as 7 is larger between 5 and seven), on the road XA such that it turns into XX₁ = X₁X₂ = X₂X₃ = X₃X₄ = X₄X₅ = X₅X₆ = X₆X₇

Step 6. Be part of the factors YX₅ and assemble a line from X₇ to YX₅ that’s parallel to the road YX₅ the place it intersects the prolonged line phase XY at level Y’.

Step 7. Now, assemble a line from Y’ the prolonged line phase XZ at Z’ that’s parallel to the road YZ, and it intersects to make a triangle.

Step 8. Therefore, ΔXY’Z’ is the wanted triangle.

Justification:

The development will be justified by proving that

XY’ = frac{7}{5}XY

Y’Z’ = frac{7}{5}YZ

XZ’= frac{7}{5}XZ

By the development, we now have Y’Z’ || YZ

Subsequently,

∠XY’Z’ = ∠XYZ {Corresponding angles}

In ΔXY’Z’ and ΔXYZ,

∠XYZ = ∠XY’Z   {As proven above}

∠YXZ = ∠Y’XZ’   {Frequent}

Subsequently,

ΔXY’Z’ ∼ ΔXYZ   { By AA similarity criterion}

Subsequently, 

frac{XY’}{XY} = frac{Y’Z’}{YZ}= frac{XZ’}{XZ}        …. (1)

In ΔXX₇Y’ and ΔXX₅Y,

∠X₇XY’=∠X₅XY (Frequent)

From the corresponding angles, we are going to get,

∠XX₇Y’=∠XX₅Y

Therefore, By the AA similarity criterion, we are going to get

ΔXX₂Y’ and XX₃Y

Thus, frac{XY’}{XY} = frac{XX_5}{XX_7}

Therefore, frac{XY }{XY’} = frac{5}{7}        ……. (2)

From the equations (1) and (2), we acquire

frac{XY’}{XY} = frac{Y’Z’}{YZ} = frac{XZ’}{ XZ} = frac{7}{5}

It may be additionally proven as

XY’ = frac{7}{5}XY

Y’Z’ =  frac{7}{5}YZ

XZ’= frac{7}{5}XZ

Thus, justified.

Query 4. Assemble an isosceles triangle whose base is 8 cm and altitude 4 cm after which one other triangle whose sides are 1frac{1}{2}      instances the corresponding sides of the isosceles triangle

Resolution:

Steps of development:

Step 1. Assemble a line phase YZ of 8 cm.

Step 2. Now assemble the perpendicular bisector of the road phase YZ and intersect on the level A.

Step 3. Taking the purpose A as centre and draw an arc with the radius of 4 cm which intersect the perpendicular bisector on the level X.

Step 4. Be part of the strains XY and XZ and the triangle is the required triangle.

Step 5. Assemble a ray YB which makes an acute angle with the road YZ on the aspect reverse to the vertex X.

Step 6. Mark the three factors Y₁, Y₂ and Y₃ on the ray YB such that YY₁ = Y₁Y₂ = Y₂Y₃

Step 7. Be part of the factors Y₂Z and assemble a line from Y₃ which is parallel to the road Y₂Z the place it intersects the prolonged line phase YZ at level Z’.

Step 8. Now, draw a line from Z’ the prolonged line phase XZ at X’, that’s parallel to the road XZ, and it intersects to make a triangle.

Step 9. Therefore, ΔX’YZ’ is the required triangle. 

Justification:

The development will be justified by proving that

X’Y = frac{3}{2}XY

YZ’ = frac{3}{2}YZ

X’Z’= frac{3}{2}XZ

By the development, we are going to acquire X’Z’ || XZ

Subsequently, 

∠ X’Z’Y = ∠XZY {Corresponding angles}

In ΔX’YZ’ and ΔXYZ,

∠Y = ∠Y (frequent)

∠X’YZ’ = ∠XZY

Subsequently,

ΔX’YZ’ ∼ ΔXYZ {By AA similarity criterion}

Therefore, 

frac{X’Y}{XY} = frac{YZ’}{YZ}= frac{X’Z}{XZ}

Thus, the corresponding sides of the same triangle are in the identical ratio, we get

frac{X’Y}{XY} = frac{YZ’}{YZ}= frac{X’Z’}{XZ} = frac{3}{2}

Thus, justified.

Query 5. Draw a triangle ABC with aspect BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then assemble a triangle whose sides are 3/4 of the corresponding sides of the triangle ABC.

Resolution:

Steps of development:

Step 1. Assemble a ΔXYZ with base aspect YZ = 6 cm, and XY = 5 cm and ∠XYZ = 60°.

Step 2. Assemble a ray YA that makes an acute angle with YZ on the other aspect of vertex X.

Step 3. Mark 4 factors (as 4 is larger in 3 and 4), comparable to Y₁, Y₂, Y₃, Y₄, on line phase YA.

Step 4. Be part of the factors Y₄Z and assemble a line via Y₃, parallel to Y₄Z intersecting the road phase YZ at Z’.

Step 5. Assemble a line via Z’ parallel to the road XZ which intersects the road XY at X’.

Step 6. Subsequently, ΔX’YZ’ is the required triangle.

Justification:

The development will be justified by proving that

Since right here the size issue is frac{3}{4}        , 

We have to show

X’Y = frac{3}{4}XY

YZ’ = frac{3}{4}YZ

X’Z’= frac{3}{4}XZ

From the development, we are going to acquire X’Z’ || XZ

In ΔX’YZ’ and ΔXYZ,

Subsequently,

∠X’Z’Y = ∠XZY {Corresponding angles}

∠Y = ∠Y {frequent}

Subsequently,

ΔX’YZ’ ∼ ΔXYZ {By AA similarity criterion}

Thus, the corresponding sides of the same triangle are in the identical ratio, we get

Subsequently,

frac{X’Y}{XY} = frac{YZ’}{YZ}= frac{X’Z’}{XZ}

Thus, it turns into 

frac{X’Y}{XY} = frac{YZ’}{YZ}= frac{X’Z’}{XZ} = frac{3}{4}

Therefore, justified.

Query 6. Draw a triangle ABC with aspect BC = 7 cm, ∠ B = 45°, ∠ A = 105°. Then, assemble a triangle whose sides are 4/3 instances the corresponding sides of ∆ ABC.

Resolution:

To search out ∠Z:

Given:

∠Y = 45°, ∠X = 105°

∠X+∠Y +∠Z = 180° {Sum of all inside angles in a triangle is 180°}

105°+45°+∠Z = 180°

∠Z = 180° − 150°

∠Z = 30°

Thus, from the property of triangle, we get ∠Z = 30°

Steps of development:

Step 1. Assemble a ΔXYZ with aspect measures of base YZ = 7 cm, ∠Y = 45°, and ∠Z = 30°.

Step 2. Assemble a ray YA makes an acute angle with YZ on the other aspect of vertex X.

Step 3. Mark 4 factors (as 4 is larger in 4 and three), comparable to Y₁, Y₂, Y₃, Y₄, on the ray YA.

Step 4. Be part of the factors Y₃Z.

Step 5. Assemble a line via Y₄ parallel to Y₃Z which intersects the prolonged line YZ at Z’.

Step 6. By means of Z’, assemble a line parallel to the road YZ that intersects the prolonged line phase at Z’.

Step 7. Therefore, ΔX’YZ’ is the required triangle.

Justification:

The development will be justified by proving that

Right here the size issue is frac{4}{3}     , we now have to show

X’Y = frac{4}{3}XY

YZ’ = frac{4}{3}YZ

X’Z’= frac{4}{3}XZ

From the development, we acquire X’Z’ || XZ

In ΔX’YZ’ and ΔXYZ,

Subsequently. 

∠X’Z’Y = ∠XZY {Corresponding angles}

∠Y = ∠Y {frequent}

Subsequently, 

ΔX’YZ’ ∼ ΔXYZ {By AA similarity criterion}

For the reason that corresponding sides of the same triangle are in the identical ratio, it turns into

Subsequently, 

frac{X’Y}{XY} = frac{YZ’}{YZ}= frac{X’Z’}{XZ}

We get,

frac{X’Y}{XY} = frac{YZ’}{YZ}= frac{X’Z’}{XZ} = frac{4}{3}        

Thus, justified.

Query 7. Draw a proper triangle by which the edges (aside from hypotenuse) are of lengths 4 cm and three cm. Then assemble one other triangle whose sides are 5/3 instances the corresponding sides of the given triangle.

Resolution:

Given:

The perimeters aside from hypotenuse are of lengths 4cm and 3cm. Therefore, the edges are perpendicular to one another.

Step of development:

Step 1. Assemble a line phase YZ =3 cm.

Step 2. Now measure and draw ∠= 90°

Step 3. Now taking Y as centre and draw an arc with the radius of 4 cm and intersects the ray on the level Y.

Step 4. Be part of the strains XZ and the triangle XYZ is the required triangle.

Step 5. Assemble a ray YA makes an acute angle with YZ on the other aspect of vertex X.

Step 6. Mark 5 comparable to Y₁, Y₂, Y₃, Y₄, on the ray YA such that YY₁ = Y₁Y₂ = Y₂Y₃= Y₃Y₄ = Y₄Y₅

Step 7. Be part of the factors Y₃Z.

Step 8. Assemble a line via Y₅ parallel to Y₃Z which intersects the prolonged line YZ at Z’.

Step 9. By means of Z’, draw a line parallel to the road XZ that intersects the prolonged line XY at X’.

Step 10. Subsequently, ΔX’YZ’ is the required triangle.

Justification:

The development will be justified by proving that

Right here the size issue is frac{5}{3}     , we have to show

X’Y = frac{5}{3}XY

YZ’ = frac{5}{3}YZ

X’Z’= frac{5}{3}XZ

From the development, we acquire X’Z’ || XZ

In ΔX’YZ’ and ΔXYZ,

Subsequently, 

∠X’Z’Y = ∠XZY {Corresponding angles}

∠Y = ∠Y {frequent}

Subsequently, 

ΔX’YZ’ ∼ ΔXYZ {By AA similarity criterion}

For the reason that corresponding sides of the same triangle are in the identical ratio, it turns into

Subsequently, 

frac{X’Y}{XY} = frac{YZ’}{YZ}= frac{X’Z’}{XZ}

So, it turns into frac{X’Y}{XY} = frac{YZ’}{YZ}= frac{X’Z’}{XZ} = frac{5}{3}

Subsequently, justified.

Constructions: Train 11.2

Query 1. Draw a circle of radius 6 cm. From a degree 10 cm away from its centre, assemble the pair of tangents to the circle and measure their lengths.

Resolution:

Building Process:

The development to attract a pair of tangents to the given circle is as follows.

Step 1. Draw a circle with radius = 6 cm with centre O.

Step 2. A degree P will be constructed 10 cm away from centre O.

Step 3. The factors O and P are then joined to kind a line

Step 4. Draw the perpendicular bisector of the road OP.

Step 5. Assemble M because the mid-point of the road phase PO.

Step 6. Utilizing M because the centre, measure the size of the road phase MO

Step 7. Now utilizing MO because the radius, draw a circle.

Step 8. The circle drawn with the radius of MO, intersect the earlier circle at level Q and R.

Step 9. Be part of the road segments PQ and PR.

Step 10. Now, PQ and PR are the required tangents.

Justification:

The development of the given downside will be justified by proving that PQ and PR are the tangents to the circle of radius 6cm with centre O. This may be proved by becoming a member of OQ and OR that are represented in dotted strains.

From the development, we will see that,

∠PQO is an angle within the semi-circle.

Each angle in a semi-circle is a proper angle, subsequently,

∴ ∠PQO = 90°

Now,

⇒ OQ ⊥ PQ

Since OQ is the radius of the circle with a radius of 6 cm, PQ have to be a tangent of the circle. Equally, we will additionally show that PR is a tangent of the circle. Therefore, justified.

Query 2. Assemble a tangent to a circle of radius 4 cm from a degree on the concentric circle of radius 6 cm and measure its size. Additionally, confirm the measurement by precise calculation.

Resolution:

Building Process:

For the desired circle, the tangent will be drawn as follows.

Step 1. Draw a circle of 4 cm radius with centre O.

Step 2. Taking O because the centre draw one other circle of radius 6 cm.

Step 3. Find a degree P on this circle

Step 4. Be part of the factors O and P to kind a line phase OP.

Step 5. Assemble the perpendicular bisector to the road OP, the place M is the mid-point

Step 6. Taking M as its centre, draw a circle with MO as its radius

Step 7. The circle drawn with the radius OM, intersect the given circle on the factors Q and R.

Step 8. Be part of the road segments PQ and PR.

Step 9. PQ and PR are the required tangents of the circle.

We will see that PQ and PR are of size 4.47 cm every.

Now, In ∆PQO,

Since PQ is a tangent,

∠PQO = 90°, PO = 6cm and QO = 4 cm

Making use of Pythagoras theorem in ∆PQO, we acquire PQ² + QO² = PQ²

PQ² + (4)² = (6)²

=> PQ² + 16 = 36

=> PQ² = 36 − 16

=> PQ² = 20

PQ = 2√5

PQ = 4.47 cm

Subsequently, the tangent size PQ = 4.47

Justification:

It may be proved that PQ and PR are the tangents to the circle of radius 4 cm with centre O.

Proof, 

Be part of OQ and OR represented in dotted strains. Now,

∠PQO is an angle within the semi-circle.

Each angle in a semi-circle is a proper angle, subsequently, ∠PQO = 90° s.t

⇒ OQ ⊥ PQ

Since OQ is the radius of the circle with a radius of 4 cm, PQ have to be a tangent of the circle. Equally, we will show that PR is a tangent of the circle.

Query 3. Draw a circle of radius 3 cm. Take two factors P and Q on certainly one of its prolonged diameters every at a distance of seven cm from its centre. Draw tangents to the circle from these two factors P and Q?

Resolution:

Building Process:

The tangent for the given circle will be constructed as follows.

Step 1. Assemble a circle with a radius of 3cm with centre O.

Step 2. Draw a diameter of a circle that extends 7 cm from the centre O of the circle and mark the endpoints as P and Q.

Step 3. Draw the perpendicular bisector of the constructed line phase PO.

Step 4. Mark the midpoint of PO as M.

Step 5. Draw one other circle with M as centre and MO as its radius

Step 6. Now be a part of the factors PA and PB by which the circle with radius MO intersects the circle of circle 3cm.

Step 7. PA and PB are the required tangents of the circle.

Step 8. From that, QC and QD are the required tangents from level Q.

Justification:

The development of the given downside will be justified by proving that PQ and PR are the tangents to the circle of radius 3 cm with centre O.

Proof, 

Be part of OA and OB. Now,

∠PAO is an angle within the semi-circle, which is the same as 90 levels.

∴ ∠PAO = 90° s.t

⇒ OA ⊥ PA

Since OA is the radius of the circle with a radius of three cm, PA have to be a tangent of the circle.

PB, QC, and QD are the tangent of the circle [by similar proof]. Therefore, justified.

Query 4. Draw a pair of tangents to a circle of radius 5 cm which is inclined to one another at an angle of 60°?

Resolution:

Building Process:

The tangents will be constructed within the following method:

Step 1. Draw a circle with a centre O of the radius of 5 cm.

Step 2. Assemble any arbitrary level Q on the circumference of the circle and be a part of the road phase OQ.

Step 3. Additionally, draw a perpendicular to QP at level Q.

Step 4. Draw a radius OR, making an angle of 120° i.e(180°−60°) with OQ.

Step 5. Draw a perpendicular to the road RP at level R.

Step 6. Each the perpendicular bisectors intersect at P.

Step 7. Subsequently, PQ and PR are the required tangents at an angle of 60°.

Justification:

The development will be justified by proving that ∠QPR = 60°

Now we have,

∠OQP = 90°, ∠ORP = 90°

Additionally ∠QOR = 120°

We all know that, the summation of the inside angles of a quadrilateral = 360°

Substituting values,

∠OQP + ∠QOR + ∠ORP + ∠QPR = 360^o

=> 90° + 120° + 90° + ∠QPR = 360°

Calculating, we get, ∠QPR = 60°

Therefore, justified.

Query 5. Draw a line phase AB of size 8 cm. Taking A as the centre, draw a circle of radius 4 cm, and taking B because the centre, draw one other circle of radius 3 cm. Assemble tangents to every circle from the centre of the opposite circle?

Resolution:

Building Process:

The tangent for the given circle will be constructed as follows.

Step 1. Assemble a line phase named AB = 8 cm.

Step 2. Taking A because the centre and draw a circle of a radius of 4 cm.

Step 3. Taking B as centre, draw one other circle of radius 3 cm.

Step 4. Draw the perpendicular bisector of the road AB with M because the midpoint.

Step 5. Taking M because the centre, draw one other circle with the radius of MA or MB which intersects the circle on the factors P, Q, R, and S.

Step 6. Be part of the road segments AR, AS, BP, and BQ respectively.

Step 7. The required tangents are AR, AS, BP, and BQ.

Justification:

The development will be justified by proving that AS and AR are the tangents of the circle with centre B and BP and BQ are the tangents of the circle with a circle centered at A.

Proof, 

Be part of AP, AQ, BS, and BR.

∠ASB is an angle within the semi-circle.

∴ ∠ASB = 90°

⇒ BS ⊥ AS

Now, BS is the radius of the circle. Subsequently, AS have to be a tangent of the circle. Equally, AR, BP, and BQ are the required tangents of the given circle.

Query 6. Let ABC be a proper triangle by which AB = 6 cm, BC = 8 cm and ∠ B = 90°. BD is perpendicular to B on AC. The circle via B, C, D is drawn. Assemble the tangents from A to this circle?

Resolution:

Building Process:

The tangent for the given circle will be constructed as follows

Step 1. Draw a line phase with base BC = 8cm

Step 2. Assemble an angle of 90° at level B, s.t ∠ B = 90°.

Step 3. Taking B as centre, draw an arc of 6cm.

Step 4. Mark the purpose of intersection as A.

Step 5. Be part of the road phase AC.

Step 6. Now, we now have ABC because the required triangle.

Step 7. Now, assemble the perpendicular bisector to the road BC with the midpoint as E.

Step 8. Taking E because the centre, draw a circle with BE or EC because the radius.

Step 9. Be part of A and E to kind a line phase.

Step 10. Now, once more draw the perpendicular bisector to the road AE and the midpoint is taken as M

Step 11. Taking M because the centre, draw a circle with AM or ME because the radius.

Step 12. Each the circles intersect at factors B and Q.

Step 13. Be part of factors A and Q to kind a line phase.

Step 14. Subsequently, AB and AQ are the required tangents

Justification:

The development will be justified by proving that AG and AB are the tangents to the circle.

Proof,

Be part of EQ.

∠AQE is an angle within the semi-circle.

∴ ∠AQE = 90°

Now, ⇒ EQ⊥ AQ

Since EQ is the radius of the circle, AQ will likely be a tangent of the circle. Additionally, ∠B = 90°

⇒ AB ⊥ BE

Since BE is the radius of the circle, AB needs to be a tangent of the circle. Therefore, justified.

Query 7. Draw a circle with the assistance of a bangle. Take a degree exterior the circle. Assemble the pair of tangents from this level to the circle?

Resolution:

Building Process:

The required tangents will be constructed on the given circle as follows.

Step 1. Draw an arbitrary circle. Mark its centre as O.

Step 2. Draw two non-parallel chords comparable to AB and CD. 

Step 3. Draw the perpendicular bisector of AB and CD

Step 4. Taking O because the centre the place each the perpendicular bisectors AB and CD intersect.

Step 5. Take a degree P exterior the circle.

Step 6. Be part of the factors O and P to kind a line phase.

Step 7. Now draw the perpendicular bisector of the road PO and mark its midpoint as M.

Step 8. Taking M as centre and MO because the radius draw a circle.

Step 9.  Each the circles intersect on the factors Q and R.

Step 10. Now be a part of PQ and PR. 

Step 11. Subsequently, PQ and PR are the required tangents.

Justification:

The development will be justified by proving that PQ and PR are the tangents to the circle. The perpendicular bisector of any chord of the circle passes via the centre. Now, be a part of the factors OQ and OR.

Each the perpendicular bisectors intersect on the centre of the circle. Since ∠PQO is an angle within the semi-circle. 

∴ ∠PQO = 90°

⇒ OQ⊥ PQ

Since OQ is the radius of the circle, PQ needs to be a tangent of the circle. Equally,

∴ ∠PRO = 90°

⇒ OR ⊥ PO

Equally, since OR is the radius of the circle, PR needs to be a tangent of the circle. Subsequently, PQ and PR are the tangents of a circle.

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Q1: Why is it necessary to study constructions?

Reply:

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Reply:

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