Class 10 Maths

Constructions (old syllabus)

14 Class 10 Maths questions on Constructions (old syllabus) from the NCERT Exemplar, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions

1. Q1 Exemplar · Exercise 10.1
   To ​divide a line ​segment $AB$ in the ​ratio $5:7$, ​first a ray $AX$ is drawn so that $\angle BAX$ is an ​acute angle and then at equal distances points are marked on ​the ray $AX$ ​such that the minimum number ​of these points is

   1. (a)8
   2. (b)10
   3. (c)11
   4. (d)12
   Show answer

   (d) 12

   Step-by-step solution →
2. Q2 Exemplar · Exercise 10.1
   To divide a line segment $AB$ in the ​ratio $4:7$, a ray $AX$ is ​drawn first such that $\angle BAX$ ​is an acute ​angle and ​then points $A_1,A_2,A_3,\dots$ are located at ​equal distances ​on the ray $AX$ and ​the point $B$ is joined to

   1. (a)$A_{12}$
   2. (b)$A_{11}$
   3. (c)$A_{10}$
   4. (d)$A_9$
   Show answer

   (b) $A_{11}$

   Step-by-step solution →
3. Q3 Exemplar · Exercise 10.1
   To divide a line ​segment ​AB in the ratio 5:6, draw ​a ray AX such ​that $\angle BAX$ ​is an ​acute angle, then draw a ray BY parallel ​to AX and the points $A_1,A_2,A_3,\dots$ and $B_1,B_2,B_3,\dots$ are located at equal distances on ray AX and BY, respectively. Then the points joined are

   1. (a)$A_5$ and $B_6$
   2. (b)$A_6$ and $B_5$
   3. (c)$A_4$ and $B_5$
   4. (d)$A_5$ and $B_4$
   Show answer

   (a) $A_5$ and $B_6$

   Step-by-step solution →
4. Q4 Exemplar · Exercise 10.1
   To ​construct a triangle similar to a ​given $△ABC$ with its sides $\frac{3}{7}$ of the corresponding sides of $△ABC$, first draw ​a ray BX such that $\angle CBX$ is an acute angle and X lies on ​the opposite side of A ​with respect to BC. ​Then locate points $B_1,B_2,B_3,\dots$ on BX at equal distances and next step ​is ​to join

   1. (a)$B_{10}$ to C
   2. (b)$B_3$ to C
   3. (c)$B_7$ to C
   4. (d)$B_4$ to C
   Show answer

   (c) $B_7$ to C

   Step-by-step solution →
5. Q5 Exemplar · Exercise 10.1
   To construct a triangle similar to a given $△ABC$ with its sides $\frac{8}{5}$ of the corresponding sides of $△ABC$, draw ​a ray BX such that $\angle CBX$ ​is an acute angle and X is ​on ​the opposite ​side of ​A with respect to BC. The minimum number of points to be located at equal distances on ​ray BX is

   1. (a)$5$
   2. (b)$8$
   3. (c)$13$
   4. (d)$3$
   Show answer

   (b) $8$

   Step-by-step solution →
6. Q6 Exemplar · Exercise 10.1
   To ​draw ​a ​pair of tangents to a circle which are inclined to each other at an angle ​of $60^{\circ }$, it ​is required to ​draw tangents at end points of those two radii ​of the circle, the angle between ​them should be

   1. (a)$135^{\circ }$
   2. (b)$90^{\circ }$
   3. (c)$60^{\circ }$
   4. (d)$120^{\circ }$
   Show answer

   (d) $120^{\circ }$

   Step-by-step solution →
7. Q7 Exemplar · Exercise 10.2
   By geometrical construction, it ​is possible to divide a line ​segment in the ratio $\sqrt{3}:\frac{1}{\sqrt{3}}$. ​State whether ​this statement ​is ​true or ​false and justify.
   Show answer

   True

   Step-by-step solution →
8. Q8 Exemplar · Exercise 10.2
   To construct a triangle similar to a given $△ABC$ with its sides $\frac{7}{3}$ ​of the corresponding ​sides of $△ABC$, ​draw ​a ray $BX$ making acute angle with $BC$ and $X$ lies on the ​opposite side of $A$ with respect to $BC$. The points $B_1,B_2,\dots ,B_7$ ​are located at equal distances on $BX$, $B_3$ is joined to $C$ and then a line segment $B_6{C}^{\prime }$ is drawn parallel to $B_{3}C$ where $C^{\prime }$ lies on $BC$ produced. Finally, line segment $A^{\prime }{C}^{\prime }$ is drawn parallel to $CA$. Is ​this construction correct? Justify.
   Show answer

   No, the construction is incorrect. Drawing the parallel through $B_6$ gives a scale factor $\frac{6}{3}=2$; for $\frac{7}{3}$ the parallel must pass through $B_7$.

   Step-by-step solution →
9. Q9 Exemplar · Exercise 10.2
   Write ​'True' or 'False' and justify your answer: A pair ​of tangents ​can ​be constructed from ​a point ​P to ​a circle of ​radius 3.5 cm situated at a distance of 3 cm from the centre.
   Show answer

   False. Since the distance $3$ cm is less than the radius $3.5$ cm, $P$ lies inside the circle and no tangent can be drawn.

   Step-by-step solution →
10. Q10 Exemplar · Exercise 10.2
    State whether ​the following ​is ​true or ​false and justify: A pair ​of tangents can be constructed ​to a circle inclined at an angle of $170^{\circ }$.
    Show answer

    True

    Step-by-step solution →
11. Q11 Exemplar · Exercise 10.3
    Draw a line ​segment of ​length 7 cm. ​Find a point P on ​it which divides it in the ​ratio 3:5.
    Show answer

    P divides AB in the ratio $3:5$, with $AP=2.625$ cm and $PB=4.375$ cm; constructed via 8 equal divisions on an auxiliary ray.

    Step-by-step solution →
12. Q12 Exemplar · Exercise 10.3
    Draw ​a ​right triangle ABC in which BC = 12 cm, AB = ​5 cm and $\angle B=90^{\circ }$. Construct a triangle similar to ​it and of ​scale ​factor $\frac{2}{3}$. ​Is the new triangle also a right triangle?
    Show answer

    Construct $△A^{\prime }B{C}^{\prime }$ with $B{C}^{\prime }=8$ cm and $A^{\prime }B=\frac{10}{3}$ cm, similar to $△ABC$ at ratio $2/3$. Yes, it is also a right triangle (right angle at $B$).

    Step-by-step solution →
13. Q13 Exemplar · Exercise 10.3
    Draw ​a ​triangle ABC in which $BC=6$ cm, $CA=5$ ​cm and $AB=4$ ​cm. ​Construct a ​triangle ​similar to it and of scale factor $\frac{5}{3}$.
    Show answer

    Triangle A'BC' similar to ABC with all sides $\frac{5}{3}$ times longer is obtained by the ray-division construction.

    Step-by-step solution →
14. Q14 Exemplar · Exercise 10.3
    Construct a ​tangent ​to a ​circle ​of radius 4 cm from a point which ​is at a distance of ​6 cm from its centre.
    Show answer

    Construct the circle on OP as diameter; it meets the given circle at the tangent points T, T'. The tangent length $PT=\sqrt{20}\approx 4.47$ cm.

    Step-by-step solution →