CBSE Class 10 Constructions Sure Shot Questions

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1. Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of 5.5 cm from its centre. [3 Marks]

Answer:
1. Draw a circle of radius 3.5 cm with centre O.
2. Take a point P at a distance of 5.5 cm from centre O and join OP.
3. Draw the perpendicular bisector of OP to locate its midpoint M.
4. Taking M as centre and MP as radius, draw a circle to intersect the given circle at two points, say Q and R.
5. Join PQ and PR. Thus, PQ and PR are the required tangents.

Teacher's Note:
a) The construction relies on the property that the angle subtended by a diameter in a semicircle is a right angle.
b) Ensure the compass is set accurately to 3.5 cm and 5.5 cm to avoid geometric errors.
c) Always label the centre O, point P, and the points of contact clearly.

2. Construct a similar \(\triangle\)ABC such that each of its side is \(\frac{2}{3}\) of the corresponding sides of \(\triangle\)ABC. It is given that AB = 5 cm, AC = 6 cm and BC = 7 cm. [4 Marks]

Answer:
1. Construct \(\triangle\)ABC with BC = 7 cm, AB = 5 cm, and AC = 6 cm.
2. Draw a ray BX making an acute angle with BC on the side opposite to vertex A.
3. Locate 3 (the greater of 2 and 3) points \(B_1\), \(B_2\), \(B_3\) on BX such that \(BB_1 = B_1B_2 = B_2B_3\).
4. Join \(B_3\) to C, and draw a line through \(B_2\) parallel to \(B_3\)C intersecting BC at C'.
5. Draw a line through C' parallel to CA intersecting AB at A'. Then \(\triangle\)A'BC' is the required similar triangle.

Teacher's Note:
a) Since the scaling factor is \(\frac{2}{3}\) (less than 1), the new triangle will be smaller than the given triangle.
b) Make sure the ray BX makes an acute angle, and use compasses to maintain equal divisions on BX.
c) Parallel line construction must be precise using set squares or a compass.
d) Verify that corresponding angles are equal by measurement.

3. Draw a line segment AB of length 4.4 cm. Taking A as centre, draw a circle of radius 2 cm and taking B as centre, draw another circle of radius 2.2 cm. Construct tangents to each circle from the centre of the other circle. [4 Marks]

Answer:
1. Draw a line segment AB = 4.4 cm.
2. With centre A and radius 2 cm, draw a circle. With centre B and radius 2.2 cm, draw another circle.
3. Find the midpoint M of AB by drawing its perpendicular bisector.
4. With centre M and radius MA (or MB), draw a circle that intersects the circle with centre A at points P and Q, and the circle with centre B at points R and S.
5. Join BP, BQ and AR, AS. These are the required tangents from B to the first circle and from A to the second circle respectively.

Teacher's Note:
a) The perpendicular bisector of the distance between the two centres is the key step to find the points of contact.
b) Students often confuse the radii of the two distinct circles; double-check compass settings.
c) Proper labelling of intersection points prevents confusion when drawing tangents.
d) The length of AB (4.4 cm) is greater than the sum of radii (2 cm + 2.2 cm = 4.2 cm), so the circles do not overlap heavily.

4. Draw a pair of tangents to a circle of radius 2 cm that are inclined to each other at an angle of 90°. [3 Marks]

Answer:
1. Draw a circle of radius 2 cm with centre O.
2. Draw any radius OA.
3. Draw a radius OB such that \(\angle\)AOB = 180° - 90° = 90°.
4. Draw perpendiculars to OA at A and to OB at B, meeting each other at point P.
5. PA and PB are the required tangents inclined to each other at 90°.

Teacher's Note:
a) The sum of angles between the radii and the angle between tangents is 180° because the quadrilateral formed is cyclic/supplementary at the radii ends.
b) A common mistake is making the angle between radii equal to the angle between tangents instead of its supplement.
c) Use a protractor or compass to construct the 90° angle accurately at the centre.

5. Draw a pair of tangents to a circle of radius 3 cm that are inclined to each other at an angle of 50°. [3 Marks]

Answer:
1. Draw a circle of radius 3 cm with centre O.
2. Draw a radius OA.
3. Construct a radius OB such that \(\angle\)AOB = 180° - 50° = 130°.
4. Draw perpendiculars to OA at point A and to OB at point B.
5. Let these perpendiculars intersect at point P. Then PA and PB are the required tangents inclined at 50°.

Teacher's Note:
a) The angle between the two radii must be 180° - 50° = 130°.
b) Precision with the protractor is essential when constructing the 130° angle.
c) Verify the final angle between the tangents using a protractor after completion.

6. Construct a tangent to a circle of radius 2 cm from a point on the concentric circle of radius 2.6 cm and measure its length. Also, verify the measurements by actual calculations. [4 Marks]

Answer:
1. Draw two concentric circles with centre O of radii 2 cm and 2.6 cm.
2. Take a point P on the outer circle.
3. Join OP and bisect it to get midpoint M.
4. With centre M and radius MP, draw a circle intersecting the inner circle at T1 and T2.
5. Join PT1 and PT2, which are the required tangents of measured length approximately 1.66 cm.
6. Calculation verification: In right-angled triangle \(OT_1P\), \(PT_1 = \sqrt{OP^2 - OT_1^2} = \sqrt{2.6^2 - 2^2} = \sqrt{6.76 - 4} = \sqrt{2.76} \approx 1.66\) cm.

Teacher's Note:
a) Pythagorean theorem is applied in the concentric circle setup to verify the tangent length.
b) Ensure geometric measurement matches the calculated algebraic value closely.
c) Label the concentric circles clearly to avoid confusion between inner and outer radii.
d) Practical ruler measurements should be within 1 mm of the calculated value.

7. Construct an isosceles triangle whose base is 7 cm and altitude 4 cm and then construct another similar triangle whose sides are \(\frac{3}{2}\) time the corresponding sides of the isosceles triangle. [4 Marks]

Answer:
1. Draw base BC = 7 cm.
2. Draw the perpendicular bisector of BC intersecting BC at D.
3. Along the perpendicular bisector, cut off DA = 4 cm. Join AB and AC to form isosceles \(\triangle\)ABC.
4. Draw a ray BX making an acute angle with BC on the side opposite to vertex A.
5. Locate 3 points \(B_1\), \(B_2\), \(B_3\) on BX at equal distances, join \(B_2\) to C, and draw a parallel line from \(B_3\) to intersect extended line BC at C'. Similarly, draw a parallel line from C' to AC intersecting extended line AB at A'. \(\triangle\)A'BC' is the required triangle.

Teacher's Note:
a) Since the scale factor is \(\frac{3}{2}\) (greater than 1), the new triangle is larger than the original triangle.
b) Sides of the triangle must be extended properly to accommodate the scaled-up version.
c) Accuracy in drawing the perpendicular bisector ensures the isosceles property.
d) Verify parallelism of lines using a set square.

8. Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are \(1\frac{1}{2}\) times the corresponding sides of the isosceles triangle. [4 Marks]

Answer:
1. Draw base BC = 8 cm.
2. Draw the perpendicular bisector of BC to find its midpoint, and mark an altitude of 4 cm upwards to locate vertex A. Join AB and AC.
3. Draw a ray BX making an acute angle with BC.
4. Note that \(1\frac{1}{2} = \frac{3}{2}\). Mark 3 equal parts on BX: \(B_1\), \(B_2\), \(B_3\).
5. Join \(B_2\) to C, draw a parallel line from \(B_3\) to meet extended BC at C', and complete the similar triangle \(\triangle\)A'BC'.

Teacher's Note:
a) Mixed fraction \(1\frac{1}{2}\) must be converted to improper fraction \(\frac{3}{2}\) before marking divisions.
b) The process is identical to scaling by a factor greater than 1.
c) Check that the base line and side AB are extended adequately.
d) Ensure clean arcs and precise intersection points.

9. Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and \(\angle\)ABC = 60°. Then construct a triangle whose sides are \(\frac{3}{4}\) of the corresponding sides of the triangle ABC. [4 Marks]

Answer:
1. Draw line segment BC = 6 cm.
2. At point B, construct an angle of 60° using a compass and draw the ray.
3. Cut off an arc of radius 5 cm from B along the ray to locate vertex A. Join AC to form \(\triangle\)ABC.
4. Draw an acute ray BX from B downwards, and mark 4 equal points \(B_1, B_2, B_3, B_4\) on it.
5. Join \(B_4\) to C, and construct a parallel line from \(B_3\) to intersect BC at C'. From C', draw a line parallel to AC to meet AB at A'. \(\triangle\)A'BC' is the required triangle.

Teacher's Note:
a) The scale factor \(\frac{3}{4}\) is less than 1, meaning the new triangle lies inside the given triangle.
b) Accurate construction of the 60° angle is vital for correct initial triangle shape.
c) Division of the ray into 4 equal segments uses standard proportional division.
d) Double-check parallel line alignments.

10. Draw a triangle ABC with side BC = 7 cm, \(\angle\)B = 45°, \(\angle\)A = 105°. Then, construct a triangle whose sides are \(\frac{4}{3}\) times the corresponding sides of \(\triangle\)ABC. [4 Marks]

Answer:
1. Calculate the missing angle \(\angle\)C = 180° - (105° + 45°) = 30°.
2. Draw base BC = 7 cm.
3. At B construct 45°, and at C construct 30°. Their intersection gives vertex A.
4. Draw an acute ray BX from B and mark 4 equal points \(B_1, B_2, B_3, B_4\).
5. Join \(B_3\) to C, draw a parallel line from \(B_4\) to meet extended BC at C', and complete the triangle by drawing a parallel line to AC from C'.

Teacher's Note:
a) Angle sum property of a triangle must be used first to find \(\angle\)C before drawing, since \(\angle\)A is given instead of \(\angle\)C.
b) Scale factor \(\frac{4}{3}\) is greater than 1, so the new triangle is larger.
c) Extension of lines BC and AB is required.
d) Verify all angle sums equal 180°.

11. Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then construct another triangle whose sides are \(\frac{5}{3}\) times the corresponding sides of the given triangle. [4 Marks]

Answer:
1. Draw a base line segment of length 4 cm and construct a 90° angle at one endpoint.
2. Cut an arc of 3 cm along the perpendicular line to form the right-angled triangle.
3. Draw an acute ray from the base vertex and divide it into 5 equal parts (since max of 5 and 3 is 5).
4. Join the 3rd division point to the base end, and draw a parallel line from the 5th division point to intersect the extended base.
5. Complete the outer similar triangle by drawing a corresponding parallel line to the hypotenuse/other side.

Teacher's Note:
a) The ratio \(\frac{5}{3}\) dictates a larger similar triangle.
b) Ensure the right angle is constructed accurately using a compass or set square.
c) Extension of the triangle's sides is necessary.
d) Verify all parallel line segments using geometric tools.

12. 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. [3 Marks]

Answer:
1. Trace a circle using a bangle on paper.
2. Draw two non-parallel chords inside the circle.
3. Draw perpendicular bisectors of these two chords; their intersection gives the centre of the circle.
4. Take a point P outside the circle and join it to the centre.
5. Draw the perpendicular bisector of the line joining the centre and P, and follow standard steps to draw tangents from an external point.

Teacher's Note:
a) A bangle gives a circle whose centre is initially unknown.
b) Chord perpendicular bisectors theorem is used to locate the hidden centre.
c) Once the centre is found, the standard tangent construction procedure applies.

13. 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. [4 Marks]

Answer:
1. Draw a circle of radius 6 cm with centre O.
2. Mark a point P at a distance of 10 cm from O, and join OP.
3. Draw the perpendicular bisector of OP to find its midpoint M.
4. With centre M and radius MP, draw a circle intersecting the given circle at T1 and T2.
5. Join PT1 and PT2 as the required tangents, and measure their lengths (measured length = 8 cm).
6. Verification: \(\sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8\) cm.

Teacher's Note:
a) Right-angled triangle relation \(\text{Tangent} = \sqrt{(\text{Distance})^2 - (\text{Radius})^2}\) confirms the length is 8 cm.
b) Precise compass work is required since dimensions are large.
c) Check that the constructed circle passes through both O and P.
d) Clearly state both graphical and calculated lengths.

14. 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. [3 Marks]

Answer:
1. Draw a circle with centre O and radius 4 cm.
2. Draw a concentric circle with centre O and radius 6 cm.
3. Take a point P on the outer circle and join OP.
4. Bisect OP to get its midpoint M, and draw a circle with centre M and radius MP intersecting the inner circle at T.
5. Join PT to form the required tangent of measured length approximately 4.47 cm.
6. Verification: In right-angled triangle OTP, OT = 4 cm and OP = 6 cm, so PT = \(\sqrt{6^2 - 4^2}\) = \(\sqrt{36 - 16}\) = \(\sqrt{20}\) = 4.47 cm.

Teacher's Note:
a) The tangent to a circle is perpendicular to the radius through the point of contact.
b) Pythagoras theorem is used in the verification step.
c) Ensure correct radii are taken for the concentric circles.

15. 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. [4 Marks]

Answer:
1. Draw a circle with centre O and radius 3 cm.
2. Draw a diameter and extend it on both sides to locate points P and Q such that OP = OQ = 7 cm.
3. Bisect OP and OQ to find their respective midpoints.
4. Draw auxiliary circles from the midpoints to intersect the main circle, and join P and Q to these intersection points to obtain the required pairs of tangents.

Teacher's Note:
a) Since points P and Q are equidistant from the centre, the lengths of tangents drawn from both points will be equal.
b) Two tangents can be drawn to a circle from an external point.
c) Accuracy in bisecting line segments is crucial for correct tangent contact points.
d) Always use sharp pencil tips for clean constructions.

16. Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°. [3 Marks]

Answer:
1. Draw a circle of radius 5 cm with centre O.
2. Draw any radius OA and construct a radius OB such that angle AOB = 120° (since 180° - 60° = 120°).
3. Draw perpendiculars to radii OA and tangents at points A and B respectively.
4. Extend these perpendiculars to intersect at an external point P, forming the required pair of tangents inclined at 60°.

Teacher's Note:
a) The angle between the two tangents and the angle between the two radii are supplementary.
b) 180° - 60° = 120° is the angle subtended at the centre.
c) Use a protractor or compass to construct the 120° angle accurately.

17. 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. [4 Marks]

Answer:
1. Draw a line segment AB = 8 cm.
2. With centre A and radius 4 cm, draw a circle. With centre B and radius 3 cm, draw another circle.
3. Bisect the line segment AB at M.
4. With centre M and radius MA (or MB), draw a circle to intersect circle A and circle B, and draw tangents from B to circle A and from A to circle B.

Teacher's Note:
a) Both sets of tangents use the common line segment AB as the distance between centres.
b) The midpoint M serves as the centre for the intersecting auxiliary circle.
c) Label all intersection points clearly to avoid confusion.
d) Double-check the initial length of AB before drawing circles.

18. Draw a circle of radius 5cm. Take a point P on it. Without using the centre of the circle, construct a tangent at the point P. Write the steps of construction also. [4 Marks]

Answer:
1. Draw a circle of radius 5 cm and take a point P on it.
2. Draw any chord PQ through point P.
3. Take any point R on the alternate segment of the circle and join PR and QR.
4. Construct an angle equal to angle PRQ at point P on the opposite side of chord PQ, forming the required tangent line.

Teacher's Note:
a) Uses the alternate segment theorem where the angle between tangent and chord equals the angle in the alternate segment.
b) No centre is required for this method.
c) Follow the alternate segment angle construction carefully.
d) Ensure the tangent line touches point P precisely.

19. Draw a circle of diameter 12cm. From a point P, 10cm away from its centre, construct a pair of tangent to the circle. Measure the lengths of the tangent segments. [3 Marks]

Answer:
1. Draw a circle of radius 6 cm (diameter 12 cm) with centre O.
2. Mark a point P at a distance of 10 cm from centre O.
3. Bisect OP and draw a reference circle intersecting the main circle at two points.
4. Join P to these points to get the tangents, each measuring approximately 8 cm in length.

Teacher's Note:
a) Radius = 6 cm, distance from centre = 10 cm.
b) Tangent length = \(\sqrt{10^2 - 6^2}\) = \(\sqrt{100 - 36}\) = \(\sqrt{64}\) = 8 cm.
c) Tangents drawn from an external point to a circle are equal in length.

20. Draw a circle of radius 5cm. from a point P, 7cm away from its centre, construct a pair of tangents to the circle. Measure the length of the tangent segments. [3 Marks]

Answer:
1. Draw a circle with centre O and radius 5 cm.
2. Take a point P at a distance of 7 cm from O.
3. Draw the perpendicular bisector of OP to find midpoint M.
4. Draw a circle with centre M and radius MO to intersect the given circle, and draw the tangents of measured length approximately 4.9 cm.

Teacher's Note:
a) Radius is 5 cm and distance is 7 cm.
b) Calculated length = \(\sqrt{7^2 - 5^2}\) = \(\sqrt{49 - 25}\) = \(\sqrt{24}\) = 4.9 cm.
c) Proper bisecting guarantees correct intersection points.

21. Draw a circle of radius 7cm. From a point P, 8cm away from its centre, Construct a pair tangents to the circle. Measure the length of the tangent segments. [3 Marks]

Answer:
1. Draw a circle with centre O and radius 7 cm.
2. Take an external point P at a distance of 8 cm from O.
3. Construct the perpendicular bisector of OP.
4. Draw the intersecting circle and join P to the intersection points, yielding tangent lengths of approximately 3.87 cm.

Teacher's Note:
a) Radius = 7 cm, distance = 8 cm.
b) Calculated length = \(\sqrt{8^2 - 7^2}\) = \(\sqrt{64 - 49}\) = \(\sqrt{15}\) = 3.87 cm.
c) Ensure clear arc intersections for accuracy.

22. Draw a right angled triangle ABC with AB = 4.5cm, AC = 7.5cm and ∠B = 90°. Construct its incircle. Write the steps of construction. [4 Marks]

Answer:
1. Draw line segment AB = 4.5 cm.
2. Construct ∠B = 90° at B, and with A as centre and radius 7.5 cm, draw an arc to intersect the ray at C, completing right-angled △ABC.
3. Draw angle bisectors of any two angles (say ∠A and ∠B) to intersect at the incentre I.
4. Drop a perpendicular from I to any side and draw the incircle.

Teacher's Note:
a) The incentre is the point of intersection of the interior angle bisectors.
b) The perpendicular distance from the incentre to any side is the inradius.
c) Use Pythagoras theorem to verify base BC = \(\sqrt{7.5^2 - 4.5^2}\) = 6 cm.
d) Proper angle bisector intersection ensures a tangent incircle.

23. Construct a triangle ABC in which BC = 13cm, CA = 5cm and AB = 12cm. Draw its incircle and measure its radius. [4 Marks]

Answer:
1. Draw base BC = 13 cm.
2. With B as centre and radius 12 cm, and C as centre and radius 5 cm, draw arcs intersecting at A; join AB and AC.
3. Draw angle bisectors of ∠B and ∠C to intersect at incentre I.
4. Drop a perpendicular from I to BC and draw the incircle, measuring its radius to be 2 cm.

Teacher's Note:
a) Notice that 5², 12², 13² form a Pythagorean triplet, making △ABC a right-angled triangle.
b) Inradius for a right-angled triangle is given by \(\frac{p + b - h}{2}\) = \(\frac{12 + 5 - 13}{2}\) = 2 cm.
c) Angle bisectors must be drawn accurately.
d) Incircle touches all three sides internally.

24. Construct a triangle ABC in which AB = 3cm, BC = 4cm and AC = 5cm. Draw the circumcircle of triangle ABC. [4 Marks]

Answer:
1. Draw BC = 4 cm.
2. With B as centre and radius 3 cm, and C as centre and radius 5 cm, draw arcs to intersect at A, completing right-angled △ABC.
3. Draw perpendicular bisectors for any two sides (say AB and BC).
4. Mark their intersection as the circumcentre, and draw the circumcircle passing through A, B, and C with radius 2.5 cm.

Teacher's Note:
a) For a right-angled triangle, the circumcentre lies at the midpoint of the hypotenuse AC.
b) The circumradius is half of the hypotenuse, which is 2.5 cm.
c) Perpendicular bisectors intersect at the circumcentre.
d) Accuracy in side measurements ensures the triangle is right-angled.

25. Construct the circumcircle of an equilateral triangle with side 6cm. Write the steps of construction. [4 Marks]

Answer:
1. Draw a line segment BC = 6 cm.
2. With B and C as centres and radius 6 cm, draw arcs intersecting at A, forming equilateral △ABC.
3. Draw perpendicular bisectors of sides AB and BC.
4. Let them intersect at circumcentre O; with O as centre and radius OB, draw the circumcircle passing through all three vertices.

Teacher's Note:
a) In an equilateral triangle, the circumcentre, incentre, centroid, and orthocentre all coincide.
b) Circumradius is given by \(\frac{\text{side}}{\sqrt{3}}\) = \(\frac{6}{\sqrt{3}}\) = \(2\sqrt{3}\) cm or approximately 3.46 cm.
c) Perpendicular bisectors intersect inside the acute triangle.
d) Steps of construction must be written clearly in sequential order.