We have provided you with Extra and Important Questions from Class 10 Maths Chapter 11 Constructions .This Extra and Important Questions will help you to score 100% in your Board Exams. These extra questions will be helpful to revise the important topics and concepts.

Table of Contents

# Constructions ** Class 10 Important Questions with Answers Maths Chapter 1**1

**Extra Questions for Class 10 Maths Chapter 11 Very Short Answer Type**

**Question:**Is construction of a triangle with sides 8 cm, 4 cm, 4 cm possible?

**Solution:**

No, we know that in a triangle sum of two sides of a triangle is greater than the third side. So the condition is not satisfied.

**Question:**To divide the line segment AB in the ratio 5 : 6, draw a ray AX such that ∠BAX is an acute angle, then draw a ray BY parallel to AX and the point A_{1}, A_{2}, A_{3}… and B_{1}, B_{2}, B_{3}… are located at equal distances on ray AX and BY respectively. Then which points should be joined?

**Solution:**

A_{5} and B_{6}.

**Question:**To draw a pair of tangents to a circle which are inclined to each other at an angle of 60°, it is required to draw tangents at end points of those two radii of the circle. What should be the angle between them?

**Solution:**

120°

**Question:**In Fig. 9.1 by what ratio does P divide AB internally.

**Solution:**

From Fig. 9.1, it is clear that there are 3 points at equal distances on AX and 4 points at equal distances on BY. Here P divides AB on joining A_{3} B_{4}. So P divides internally by 3 : 4.

## Extra Questions for Class 10 Maths Chapter 11 Short Answer Type

**Question:**

Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are 23 of the corresponding sides of the first triangle.

OR

Draw a triangle with sides 4 cm, 5 cm and 6 cm. Then construct another triangle whose sides are 23 of the corresponding sides of first triangle.

**Solution:**

Steps of Construction:

Step I: Draw a line segment BC = 6 cm

Step II: Draw an arc with B as centre and radius equal to 5 cm.

Step III: Draw an arc, with C as centre and radius equal to 4 cm intersecting the previous drawn arc at A.

Step IV: Join AB and AC, then ∆ABC is the required triangle.

Step V: Below BC make an acute angle CBX

Step VI: Along BX mark off three points at equal distance: B_{1}, B_{2}, B_{3}, such that BB_{1} = B_{1}B_{2}, = B_{2}B_{3}.

Step VII: Join BC_{3}.

Step VIII: From B_{2}, draw B_{2}, D || B_{3},C, meeting BC at D.

Step IX: From D draw ED || AC meeting BA at E. Then we have ∆EDB which is the required triangle.

Justification:

Since DE || CA

Hence, we have the new AEBD similar to the given ∆ABC, whose sides are equal to 23 of the corresponding sides of ∆ABC.

**Question:**Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts.

**Solution:**

Steps of Construction:

Step I: Draw a line segment AB = 7.6 cm

Step II: Draw any ray AX making an acute angle ∠BAX with AB.

Step III: On ray AX starting from A, mark 5 + 8 = 13 equal arcs. AA_{1}, A_{1}A_{2}, A_{2}A_{3}, A_{3}A_{4}, … A_{11}A_{12}, and A_{12}A_{13}.

Step IV: Join A_{13}B.

Step V: From A_{5}, draw A_{5}P || A_{13}B, meeting AB at P. Thus, P divides AB in the ratio 5 : 8. On measuring the two parts. We find AP = 2.9 cm and PB = 4.7 cm (approx).

Justification:

In ∆ABA_{13}, PA_{5} || BA_{13} .

∴ By Basic Proportionality Theorem

**Question:**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:**

Steps of Construction:

Step 1: Take a point O and draw a circle of radius 6 cm.

Step II: Take a point P at a distance of 10 cm from the centre 0.

Step III: Join OP and bisect it. Let M be the mid-point.

Step IV: With M as centre and MP as radius, draw a circle to intersect the circle at Q and R.

Step V: Join PQ and PR. Then, PQ and PR are the required tangents. On measuring, we find, PQ = PR = 8cm.

Justification:

On joining OQ, we find that ∠PQO = 90°, as ∠PQO is the angle in the Semicircle.

∴ PQ ⊥ OQ

Since OQ is the radius of the given circle, PQ has to be a tangent to the circle. Similarly, PR is

also a tangent to the circle

**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:**

Steps of Construction:

Step 1: Take a point O 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 BP = BQ = 4.5 cm.

Justification:

In ∆BPO, we have

∠BPO = 90°, OB = 6 cm and OP = 4 cm

∴ OB^{2} = BP^{2} + OP^{2} [Using Pythagoras theorem]

Similarly, BQ = 4.47 cm

**Question:**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.

**Solution:**

Steps of Construction:

Step I: Draw a line segment AB = 8 cm.

Step II: With A as centre, draw a circle of radius 4 cm and let it intersect the line segment AB in M.

Step III: With B as centre, draw a circle of radius 3 cm.

Step IV: With M as centre, draw a circle of radius AM and let it intersect the given two circles in P, e and R, S.

Step V: Join AP, AQ, BR and BS.

These are the required tangents.

Justification:

On joining BP, we have ∠BPA = 90°, as ∠BPA is the angle in the semicircle.

∴ AP ⊥ PB

Since BP is the radius of given circle, so AP has to be a tangent to the circle. Similarly, AQ, BR and BS are the tangents.

**Question:**Draw a circle of radius of 3 cm. Take two points P and Q on one of its diameters extended on both sides, each at a distance of 7 cm on opposite sides of its centre. Draw tangents to the circle from these two points P and Q.

**Solution:**

Steps of Construction:

Step 1: Taking a point ( as centre, draw a circle of radius 3 cm.

Step II: Take two points P and Q on one of its extended diameter such that OP = OQ = 7 cm.

Step III: Bisect OP and OQ and let M_{1} and M_{2} be the mid-points of OP and OQ respectively.

Step IV: Draw a circle with M_{1} as centre and M_{1} P as radius to intersect the circle at T_{1}, and T_{2}.

Step V: Join PT_{1} and PT_{2}.

Then, PT_{1} and PT_{2} are the required tangents. Similarly, the tangents QT_{3} and QT_{4} can be obtained

Justification:

On joining OT_{1}, we find ∠PT_{1}O = 90°, as it is an angle in the semicircle.

PT_{1} ⊥ OT_{1}

Since OT_{1} is a radius of the given circle, so PT_{1} has to be a tangent to the circle.

Similarly, PT_{2}, QT_{3} and QT_{4} are also tangents to the circle.

**Question:**Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and ∠B = 90°. BD is the perpendicular from B on AC. The circle through B, C, D is drawn. Construct the tangents from A to this circle.

**Solution:**

Steps of Construction:

Step I: Draw ∆ABC and perpendicular BD from B on AC.

Step II: Draw a circle with BC as a diameter. This circle will pass through D.

Step III: Let O be the mid-point of BC. Join A0.

Step IV: Draw a circle with AO as diameter. This circle cuts the circle drawn in step II at B and E.

Step V: Join AE. AE and AB are desired tangents drawn from A to the circle passing through B, C and D.

## Extra Questions for Class 10 Maths Chapter 11 Long Answer Type

**Question:**Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circle.

**Solution:**

Steps of Construction:

Step I: Draw a circle with the help of a bangle.

Step II: Let P be the external point from where the tangents are to be drawn to the given circle. Through P, draw a secant PAB to intersect the circle at A and B (say).

Step III: Produce AP to a point C, such that AP = PC, i.e., P, is the mid-point of AC.

Step IV: Draw a semicircle with BC as diameter.

Step V: Draw PD ⊥ CB, intersecting the semicircle at D.

Step VI: With P as centre and PD as radius, draw arcs to intersect the given circle at T and T_{1}.

Step VII: Join PT and PT_{1}. Then, PT and PT_{1} are the required tangents.

**Question:**Draw a pair of tangents to a circle of radius 4 cm which are inclined to each other at an angle of 60°

OR

Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is 60°. Also justify the construction. Measure the distance between the centre of the circle and the point of intersection of tangents.

**Solution:**

Steps of Construction:

Step I: Draw a circle with centre 0 and radius 4 cm.

Step II: Draw any diameter AOB.

Step III: Draw a radius OC such that ∠BOC = 60°.

Step IV: At C, we draw CM ⊥ OC and at A, we draw AN ⊥ OA.

Step V: Let the two perpendiculars intersect each other at P. Then, PA and PC are required tangents.

Justification:

Since OA is the radius, so PA has to be a tangent to the circle. Similarly, PC is also tangent to the circle.

∠APC = 360° – (∠OAP + ∠OCP + ∠AOC)

= 360° – (90° + 90° + 120°) = 360° – 300° = 60°

Hence, tangents PA and PC are inclined to each other at an angle of 60°

**Question:**Draw an isosceles ΔABC in which BC = 5.5 cm and altitude AL = 3 cm. Then construct another triangle whose sides are 3/4 of the corresponding sides of ΔABC

**Solution:**

Steps of construction:

- Draw BC = 5.5 cm.
- Construct AP the perpendicular bisector of BC meeting BC at L.
- Along LP cut off LA = 3 cm.
- Join BA and CA. Then ΔABC so obtained is the required ΔABC.
- Draw an acute angle CBY and cut 4 equal lengths as BA1 = A1A2 = A2A3 = A3A4 and join CA4.
- Now draw a line through A3 parallel to CA4 intersecting BC at C’.
- Draw a line through C’ and parallel to AC intersecting AB at A’. BA’C’ is the required triangle.

**Question:**Draw a triangle with sides 5 cm, 6 cm and 7 cm. Then draw another triangle whose 4/5 sides are y of the corresponding sides of first triangle

**Solution:**

- Draw a line segment AB of length 7 cm.

Then using A as centre and distance 5 cm draw an arc C.

Also draw an arc using B as centre and with distance 6 cm, which intersect earlier drawn arc at C. Join AC and BC. - Draw an acute angle BAZ and cut AZ as AA1 = A1A2 = A2A3 = A3A4 = A4A5 and join BA5.
- Through A4 draw a line parallel BA5 intersecting AB at B’.
- Through B’ draw a line parallel to BC intersecting AC at C’. AAB’C’ is the required triangle.

**Question:**Draw a ΔABC in which AB = 4 cm, BC = 5 cm and AC = 6 cm. Then construct another triangle whose sides are 3/5 of the corresponding sides of ΔABC

**Solution:**

Steps of construction:

- Draw a line segment AC = 6 cm.
- Draw an arc with A as centre and radius equal to 5 cm.
- Draw an arc with C as centre and radius equal to 4 cm intersecting the previous drawn arc at B.
- Join AB and CB, then ΔABC is required triangle.
- Below AC make an acute angle CAX.
- Along AX mark of 5 points A1, A2, A3, A4, A5 such that AA1= A1A2 = A2A3= A3A4 = A4A5.
- Join A5C.
- From A3 draw A3D | | A3C meeting AC at D.
- From D, draw ED | | BC meeting AB at E. Then we have ΔEDA which is the required triangle.