CLASS X : CHAPTER - 11 CONSTRUCTIONS

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CLASS X : CHAPTER - 11 CONSTRUCTIONS

IMPORTANT CONCEPTS

1. To construct a triangle similar to a given triangle as per given scale factor:
This involves creating a triangle where sides are a fraction (scale factor) of the corresponding sides of the original triangle.

Example: Scale Factor \( \frac{3}{4} \) (Scale factor < 1)
To construct a triangle similar to \( \Delta ABC \) with sides equal to \( \frac{3}{4} \) of the corresponding sides:

• Draw a ray BX making an acute angle with BC on the side opposite to vertex A.
• Locate 4 points (the greater of 3 and 4) \( B_1, B_2, B_3, B_4 \) on BX at equal distances.
• Join \( B_4C \) and draw a line through \( B_3 \) parallel to \( B_4C \) to intersect BC at \( C' \).
• Draw a line through \( C' \) parallel to CA to intersect BA at \( A' \). Then \( \Delta A'BC' \) is the required triangle.

Example: Scale Factor \( \frac{5}{3} \) (Scale factor > 1)
To construct a triangle similar to \( \Delta ABC \) with sides equal to \( \frac{5}{3} \) of the corresponding sides:

• Draw a ray BX making an acute angle with BC.
• Locate 5 points (the greater of 5 and 3) \( B_1, \dots, B_5 \) on BX at equal distances.
• Join \( B_3 \) (the smaller number in the ratio) to C.
• Draw a line through \( B_5 \) parallel to \( B_3C \), 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 \( \Delta A'BC' \) is the required triangle.

2. To construct the tangents to a circle from a point outside it:
Given a circle with centre 'O' and a point P outside it:

• Join PO and draw a perpendicular bisector of it. Let M be the midpoint of PO.
• Taking M as centre and PM (or MO) as radius, draw a circle. Let it intersect the given circle at points A and B.
• Join PA and PB. These are the required tangents.

3. To construct a tangent to a circle at a given point when the centre is known:

• Draw a circle with centre 'O' and mark a point 'P' on it. Join OP.
• Draw a perpendicular line through the point P. This line is the required tangent.

MCQ WORKSHEET-I

1. To divide a line segment AB in the ratio 3 : 7, first a ray AX is drawn so that angle BAX is an acute angle and then at equal distances point are marked on the ray AX such that the minimum number of these point is
   1. 3
   2. 10
   3. 7
   4. 12
2. To divide a line segment AB in the ratio 4 : 5, first 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. \( A_4 \)
   2. \( A_5 \)
   3. \( A_{10} \)
   4. \( A_9 \)
3. To construct a triangle similar to a given \( \Delta ABC \) with its sides \( \frac{2}{5} \) of the corresponding sides of \( \Delta ABC \), first draw a ray BX... The minimum number of points to be located at equal distances on ray BX is
   1. 3
   2. 5
   3. 8
   4. 2
4. To draw a pair of tangents to a circle which are inclined to each other at an angle of \( 30^\circ \), it is required to draw tangents at end points of those two radii of the circle, the angle between them, should be
   1. \( 150^\circ \)
   2. \( 90^\circ \)
   3. \( 60^\circ \)
   4. \( 120^\circ \)
5. To draw the perpendicular bisector of line segment AB, we open the compass
   1. more than \( \frac{1}{2} AB \)
   2. less than \( \frac{1}{2} AB \)
   3. equal to \( \frac{1}{2} AB \)
   4. none of these
6. To construct a triangle we must know at least its ______ parts.
   1. two
   2. three
   3. one
   4. five
7. Construction of a triangle is not possible if:
   1. \( AB + BC < AC \)
   2. \( AB + BC = AC \)
   3. both (a) and (b)
   4. \( AB + BC > AC \)
8. With the help of ruler and compass it is not possible to construct an angle of
   1. \( 37.5^\circ \)
   2. \( 40.5^\circ \)
   3. \( 22.5^\circ \)
   4. \( 67.5^\circ \)
9. The construction of a triangle ABC given that \( BC = 3 \) cm, \( \angle C = 60^\circ \) is possible when difference of AB and AC is equal to
   1. 3.2 cm
   2. 3.1 cm
   3. 3 cm
   4. 2.8 cm

PRACTICE QUESTIONS

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.
2. Construct a similar \( \Delta ABC \) such that each of its side is \( \frac{2}{3} \) of the corresponding sides of \( \Delta ABC \). It is given that AB = 5 cm, AC = 6cm and BC = 7cm.
3. Draw a line segment AB of length 4.4cm. Taking A as centre, draw a circle of radius 2cm and taking B as centre, draw another circle of radius 2.2cm. Construct tangents to each circle from the centre of the other circle.
4. Draw a pair of tangents to a circle of radius 2 cm that are inclined to each other at an angle of \( 90^\circ \).
5. Construct a tangent to a circle of radius 2 cm from a point on the concentric circle of radius 2.6cm and measure its length. Also, verify the measurements by actual calculations.
6. Construct an isosceles triangle whose base is 7 cm and altitude 4 cm and then construct another similar triangle whose sides are \( \frac{3}{2} \) times the corresponding sides of the isosceles triangle.
7. Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and \( \angle ABC = 60^\circ \). Then construct a triangle whose sides are \( \frac{3}{4} \) of the corresponding sides of the triangle ABC.
8. 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.
9. 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.
10. 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.
11. 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.
12. Draw a right angled triangle ABC with AB = 4.5cm, AC = 7.5cm and \( \angle B = 90^\circ \). Construct its incircle. Write the steps of construction.
13. Construct a triangle ABC in which BC = 13cm, CA = 5cm and AB = 12cm. Draw its incircle and measure its radius.
14. Construct a triangle ABC in which AB = 3cm, BC = 4cm and AC = 5cm. Draw the circumcircle of triangle ABC.
15. Construct the circumcircle of an equilateral triangle with side 6cm. Write the steps of construction.

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