CONSTRUCTIONS - Q&A
EXERCISE 181. Multiple Choice Type: Choose the correct answer from the options given below.
(i) In a quadrilateral ABCD, side AB will be parallel to side DC, if:
(a) ∠A + ∠B = 180°
(b) ∠A = ∠B
(c) ∠D = ∠C
(d) ∠B + ∠C = 180°
Answer: (d) ∠B + ∠C = 180°
Explanation: For lines AB and DC to be parallel, the consecutive interior angles (co-interior angles) between them must be supplementary. Here, ∠B and ∠C are the interior angles on the same side of the transversal BC.
(ii) In which of the case, out of the following cases, it is not possible to construct a quadrilateral; when:
A. all the four sides are given.
B. one diagonal and all the four sides are given.
C. one angle and all the four sides are given.
D. both the diagonals and three sides are given.
Answer: A. all the four sides are given.
Explanation: Knowing only the four sides is not sufficient to construct a unique quadrilateral because the shape is not rigid (it can be deformed). We need at least 5 independent measurements (like a diagonal or an angle) to fix the shape.
(iii) In which of the case, out of the following cases, it is not possible to construct a parallelogram, when:
A. two adjacent sides and angle between them are given.
B. two adjacent sides and a diagonal are given.
C. both the diagonals are given.
D. both the diagonals and the angle between them are given.
Answer: C. both the diagonals are given.
Explanation: To construct a unique parallelogram, knowing just the lengths of the diagonals is insufficient. We also need the angle between them or the side lengths to determine the exact shape.
(iv) In which of the following cases, it is not possible to construct a rhombus when:
A. both the diagonals are given.
B. one side and one diagonal are given.
C. one side and one angle are given.
D. one diagonal is given.
Answer: D. one diagonal is given.
Explanation: A single diagonal does not define a unique rhombus. Infinite rhombuses can be drawn with a single fixed diagonal length.
(v) In which of the following cases, it is not possible to construct a square when:
A. one diagonal is given
B. one side is given
C. one angle is given which is 90°
D. one side and the angle between a side and a diagonal are given.
Answer: C. one angle is given which is 90°
Explanation: By definition, all squares have 90° angles. Just knowing an angle is 90° does not tell us the size (side length) of the square.
2. Given below are the angles x and y. Without measuring these angles, construct:
(i) ∠ABC = x + y
(ii) ∠ABC = 2x + y
(iii) ∠ABC = x + 2y
Steps of Construction:
1. Draw a ray BC.
2. Place the compass at the vertex of the given angle x and draw an arc cutting both arms. With the same radius, draw an arc from point B cutting ray BC.
3. Measure the distance between the arms of angle x with the compass and cut this length on the arc drawn from B. This constructs angle x.
4. From the end of angle x, repeat the process to add angle y adjacent to it.
5. For (ii), add angle x, then angle x again, then angle y.
6. For (iii), add angle x, then angle y, then angle y again. The total angle formed is the required ∠ABC.
3. Draw angle ABC of any suitable measure.
(i) Draw BP, the bisector of angle ABC.
(ii) Draw BR, the bisector of angle PBC and draw BQ, the bisector of angle ABP.
(iii) Are the angles ABQ, QBP, PBR and RBC equal?
(iv) Are the angles ABR and QBC equal?
Steps of Construction:
1. Draw an angle ∠ABC (e.g., 80°).
2. With B as center, draw an arc cutting AB and BC. From these intersection points, draw intersecting arcs to find the point for the bisector. Join B to this point to get BP.
3. Similarly, bisect ∠PBC to get BR, and bisect ∠ABP to get BQ.
Answers:
(iii) Yes. BP divides the main angle into two equal halves. BQ and BR divide those halves into equal quarters. Therefore, ∠ABQ = ∠QBP = ∠PBR = ∠RBC.
(iv) Yes. ∠ABR consists of three of these equal parts (quarters), and ∠QBC also consists of three equal parts. Thus, they are equal.
4. In each of the following, draw a perpendicular through point P to the line segment AB :
(i) P is on the line AB.
(ii) P is above the line AB.
(iii) P is at the end of line AB (or slightly extended).
Steps of Construction:
(i) P on line: Place compass at P, draw arcs cutting AB on both sides. From these points, draw arcs of equal radius above P to intersect. Join P to the intersection.
(ii) P above line: Place compass at P, draw an arc cutting line AB at two points. From these two points, draw arcs below the line to intersect. Join P to this intersection.
(iii) P at end: Extend the line AB if needed. Use the same method as (i) or construct a 90° angle using the standard semi-circle method.
5. Draw a line segment AB = 5.5 cm. Mark a point P, such that PA = 6 cm and PB = 4.8 cm. From the point P, draw a perpendicular to AB.
Steps of Construction:
1. Draw line segment AB = 5.5 cm.
2. From A, draw an arc of radius 6 cm. From B, draw an arc of radius 4.8 cm. The intersection of these arcs is point P.
3. Place compass at P and draw an arc that cuts AB at two points (extend AB if necessary).
4. From these two points on the line, draw intersecting arcs on the opposite side of the line.
5. Join P to this intersection point to get the perpendicular.
6. Draw a line AB = 6 cm. Mark a point P any where outside the line AB. Through the point P, construct a line parallel to AB.
Steps of Construction:
1. Draw AB = 6 cm. Mark point P outside.
2. Take any point Q on AB and join PQ.
3. At point P, construct an angle equal to ∠PQA (alternate interior angles) on the opposite side of PQ.
4. Extend the arm of this new angle to draw the line parallel to AB.
7. Draw a line MN = 5.8 cm. Locate a point A which is 4.5 cm from M and 5 cm from N. Through A draw a line parallel to line MN.
Steps of Construction:
1. Draw MN = 5.8 cm.
2. With M as center and radius 4.5 cm, draw an arc. With N as center and radius 5 cm, draw an arc. Their intersection is point A.
3. Take a point Q on MN and join AQ.
4. Construct an angle at A equal to ∠AQN. The line through A making this angle is parallel to MN.
8. Draw a straight line AB = 6.5 cm. Draw another line which is parallel to AB at a distance of 2.8 cm from it.
Steps of Construction:
1. Draw AB = 6.5 cm.
2. Mark two points on AB. At these points, construct perpendiculars to AB.
3. Using a compass set to 2.8 cm, cut off segments on these perpendiculars.
4. Join the points obtained. This line is parallel to AB at a distance of 2.8 cm.
9. Construct an angle PQR = 60°. Draw a line parallel to PQ at a distance of 3 cm from it and another line parallel to QR at a distance of 3.5 cm from it. Mark the point of intersection of these parallel lines as A.
Steps of Construction:
1. Construct ∠PQR = 60°.
2. To draw parallel to PQ: Erect perpendiculars on PQ, cut off 3 cm, and join the points.
3. To draw parallel to QR: Erect perpendiculars on QR, cut off 3.5 cm, and join the points.
4. Extend these two newly constructed parallel lines until they intersect. Mark this point as A.
10. Draw an angle ABC = 60°. Draw the bisector of it. Also draw a line parallel to BC a distance of 2.5 cm from it. Let this parallel line meet AB at point P and angle bisector at point Q. Measure the lengths of BP and PQ. Is BP = PQ?
Steps of Construction:
1. Construct ∠ABC = 60° and draw its bisector BD.
2. Construct a line parallel to BC at a distance of 2.5 cm.
3. Let this parallel line intersect AB at P and the bisector at Q.
4. Measure BP and PQ.
Answer: Yes, BP = PQ.
Explanation: Since PQ is parallel to BC, alternate angles ∠PQB and ∠QBC are equal. Since BQ is a bisector, ∠PBQ = ∠QBC. Therefore, ∠PQB = ∠PBQ, making △PBQ an isosceles triangle.
11. Construct a quadrilateral ABCD; if :
(i) AB = 8 cm, BC = 5.4 cm, AD = 6 cm, ∠A = 60° and ∠B = 75°
(ii) AB = 6 cm = AC, BC = 4 cm, CD = 5 cm and AD = 4.5 cm.
(iii) AB = AD = 5 cm, BD = 7 cm and BC = DC = 5.5 cm.
Steps of Construction:
(i) Draw AB=8 cm. At A construct 60°, cut AD=6 cm. At B construct 75°, cut BC=5.4 cm. Join CD.
(ii) Draw AB=6 cm. From A arc 6 cm (AC), from B arc 4 cm (BC) → intersect at C. From A arc 4.5 cm (AD), from C arc 5 cm (CD) → intersect at D. Join all points.
(iii) Draw BD=7 cm. From B and D, draw arcs of 5 cm above BD to find A. From B and D, draw arcs of 5.5 cm below BD to find C. Join vertices.
12. Construct a parallelogram ABCD, if :
(i) AB = 3.6 cm, BC = 4.5 cm and ∠ABC = 120°
(ii) BC = 4.5 cm, CD = 5.2 cm and ∠ADC = 60°
(iii) AD = 4 cm, DC = 5 cm and diagonal BD = 7 cm.
(iv) diagonal AC = 6.4 cm, diagonal BD = 5.6 cm and angle between the diagonals is 75°
Steps of Construction:
(i) Draw AB=3.6 cm. At B, angle 120°. Cut BC=4.5 cm. Complete parallelogram by drawing arcs from A (radius 4.5) and C (radius 3.6) to meet at D.
(ii) Draw CD=5.2 cm. At D, angle 60°. Cut DA=4.5 cm (since opposite sides equal). Complete parallelogram.
(iii) Construct △ABD with sides AD=4, AB=5 (since AB=DC), BD=7. From B and D, draw remaining sides to find C.
(iv) Draw AC=6.4 cm. Mark midpoint O. Draw line through O at 75°. Cut OB = OD = 2.8 cm (half of 5.6). Join vertices.
13. Construct a rectangle ABCD; if:
(i) AB = 4.5 cm and BC = 5.5 cm.
(ii) AB = 5.0 cm and diagonal AC = 6.7 cm.
(iii) each diagonal is 6 cm and the angle between them is 45°
Steps of Construction:
(i) Draw AB=4.5 cm. Construct 90° at B. Cut BC=5.5 cm. Complete the rectangle using equal opposite sides.
(ii) Draw AB=5 cm. Construct 90° at B. From A, draw arc of radius 6.7 cm to cut the perpendicular at C. Complete the rectangle.
(iii) Draw diagonal AC=6 cm. Find midpoint O. Draw line at 45° through O. Cut lengths of 3 cm (half of 6) on both sides to find B and D. Join vertices.
14. Construct a rhombus ABCD, if:
(i) AB = 4 cm and ∠B = 120°
(ii) CD = 5 cm and diagonal BD = 8.5 cm.
(iii) BC = 4.8 cm and diagonal AC = 7 cm.
(iv) diagonal AC = 6.6 cm and diagonal BD = 5.8 cm.
Steps of Construction:
(i) Draw AB=4 cm. At B, angle 120°. Cut BC=4 cm. From A and C, draw arcs of 4 cm to find D.
(ii) All sides = 5 cm. Construct △BCD with sides 5, 5, 8.5. From B and D, draw arcs of 5 cm to find A.
(iii) All sides = 4.8 cm. Construct △ABC with sides 4.8, 4.8, 7. From A and C, draw arcs of 4.8 cm to find D.
(iv) Draw AC=6.6 cm. Construct perpendicular bisector. Cut 2.9 cm (half of 5.8) above and below to find B and D. Join vertices.
15. Using ruler and compasses only, construct a parallelogram ABCD, in which: AB = 6 cm, AD = 3 cm and ∠DAB = 60°. In the same figure draw the bisector of angle DAB and let it meet DC at point P. Measure angle APB.
Steps of Construction:
1. Draw AB = 6 cm.
2. Construct ∠A = 60°. Cut AD = 3 cm.
3. From D, draw arc radius 6 cm. From B, draw arc radius 3 cm. Intersect at C. Join to form parallelogram.
4. Construct bisector of ∠A. Extend it to meet DC at P.
5. Measure ∠APB.
Answer: ∠APB = 90°.
Explanation: ∠DAP = 30° (bisector). ∠D = 120° (consecutive angles). In △ADP, ∠DPA = 180 - (120+30) = 30°. Thus AD=DP=3. Since DC=6, P is midpoint. △BPC is equilateral (sides 3,3, angle 60). ∠BPC=60°. Angle on line at P = 30° + ∠APB + 60° = 180°. So ∠APB = 90°.
Test yourself
1. Multiple Choice Type: Choose the correct answer from the options given below.
(i) Let angle ABC = 60° and angle ABD = 90° then ∠CBD is equal to:
(a) 150°
(b) 30°
(c) 150° or 30°
(d) none of these
Answer: (c) 150° or 30°
Explanation: If D and C are on opposite sides of AB, angle is 90+60=150°. If on the same side, angle is 90-60=30°.
(ii) The shortest distance between the point P and the line segment AB is:
(a) PA
(b) PB
(c) line joining point P with the mid-point of AB.
(d) length of perpendicular from point P to line AB.
Answer: (d) length of perpendicular from point P to line AB.
(iii) Two lines AB and CD are not parallel to each other, when they are cut by a transversal with:
(a) vertically opposite angles equal.
(b) corresponding angles equal
(c) alternate angles equal
(d) co-interior angles are supplementary.
Answer: (a) vertically opposite angles equal.
Explanation: Vertically opposite angles are always equal regardless of whether lines are parallel or not. The other options are conditions that prove lines are parallel.
(iv) A parallelogram will not be a rhombus if:
(a) sum of its opposite angles is 180°
(b) its adjacent sides are equal.
(c) its diagonals are perpendicular to each other.
(d) each diagonal bisects the angles of the vertices.
Answer: (a) sum of its opposite angles is 180°
Explanation: If opposite angles sum to 180° in a parallelogram (where opposite angles are already equal), each angle is 90°. This makes it a rectangle. A rectangle is not necessarily a rhombus.
(v) A rectangle will not be a square, if:
(a) adjacent sides are equal
(b) angle between the diagonals is 90°
(c) diagonals bisect each other and the angle between them is not 90°
(d) all its sides are equal.
Answer: (c) diagonals bisect each other and the angle between them is not 90°
Explanation: For a rectangle to become a square, the diagonals must intersect at 90°. If they don't, it remains a rectangle.
2. Given below are the angles x, y and z. Without measuring these angles construct:
(i) ∠ABC = x + y + z
(ii) ∠ABC = 2x + y + z
(iii) ∠ABC = x + 2y + z
Steps of Construction:
Draw a base ray. Using a compass, copy angle x onto the ray. From the new arm, copy angle y. From that arm, copy angle z. The total opening is the sum. For 2x, copy x twice sequentially.
3. Draw a line segment AB = 6.2 cm. Mark a point P in AB such that BP = 4 cm. Through point P draw a perpendicular to AB.
Steps of Construction:
Draw AB=6.2 cm. Measure 4 cm from B to find P. At P, construct a perpendicular using the semi-circle and intersecting arcs method.
4. Construct an angle ABC = 90°. Locate a point P which is 2.5 cm from AB and 3.2 cm from BC.
Steps of Construction:
Construct ∠ABC=90°. Draw a line parallel to AB at 2.5 cm distance. Draw a line parallel to BC at 3.2 cm distance. Their intersection is point P.
5. Construct a quadrilateral ABCD; if :
(i) AB = 4.3 cm, BC = 5.4 cm, CD = 5 cm, DA = 4.8 cm and angle ABC = 75°.
(ii) AB = 6 cm, CD = 4.5 cm, BC = AD = 5 cm and ∠BCD = 60°.
(iii) AB = 5 cm, BC = 6.5 cm, CD = 4.8 cm, ∠B = 75° and ∠C = 120°
Steps of Construction:
(i) Draw BC. At B, angle 75°. Cut AB=4.3. From A, arc 4.8 (DA). From C, arc 5 (CD). Intersect at D.
(ii) Draw BC=5. At C, angle 60°. Cut CD=4.5. D is found. From D, arc 5 cm (AD). From B, arc 6 cm (AB). Intersect at A. Join vertices.
(iii) Draw BC=6.5. At B, angle 75°. At C, angle 120°. Cut AB=5 and CD=4.8. Join A and D.
6. Construct a parallelogram ABCD, if :
(i) AB = 5.8 cm, AD = 4.6 cm and diagonal AC = 7.5 cm.
(ii) lengths of diagonals AC and BD are 6.3 cm and 7.0 cm respectively, and the angle between them is 45°.
(iii) lengths of diagonals AC and BD are 5.4 cm and 6.7 cm respectively, and the angle between them is 60°.
Steps of Construction:
(i) Construct △ABC (AB=5.8, BC=4.6, AC=7.5). From A and C, draw arcs of 4.6 and 5.8 to find D.
(ii) Draw AC=6.3. Midpoint O. Line at 45°. Cut OB=OD=3.5. Join vertices.
(iii) Same as (ii) with lengths 5.4 (half 2.7) and 6.7 (half 3.35) and angle 60°.
7. Construct a rectangle ABCD; if :
(i) BC = 6.1 cm and CD = 6.8 cm.
(ii) AD = 4.8 cm and diagonal AC = 6.4 cm.
(iii) each diagonal is 5.5 cm and the angle between them is 60°.
Steps of Construction:
(i) Standard rectangle construction with sides 6.1 and 6.8.
(ii) Draw AD=4.8. Perpendicular at D. Arc from A (radius 6.4) cuts perpendicular at C. Complete rectangle.
(iii) Draw diagonal AC=5.5. Midpoint O. Line at 60°. Cut 2.75 cm on both sides for B and D. Join vertices.
8. Construct a rhombus ABCD, if :
(i) BC = 4.7 cm and ∠B = 75°
(ii) diagonal AC = 4.9 cm and diagonal BD = 6 cm.
(iii) diagonal AC = 8 cm and diagonal BD = 6 cm.
Steps of Construction:
(i) Draw BC=4.7. Angle 75° at B. Cut AB=4.7. Complete rhombus using equal sides.
(ii) Draw BD=6. Perpendicular bisector. Cut 2.45 cm (half of 4.9) above and below.
(iii) Draw AC=8. Perpendicular bisector. Cut 3 cm (half of 6) above and below.
9. Construct a quadrilateral ABCD in which ∠A = 120° ∠B = 60°, AB = 4 cm, BC = 4.5 cm and CD = 5 cm.
Steps of Construction:
1. Draw AB = 4 cm.
2. Construct ∠A = 120° and ∠B = 60°.
3. Cut BC = 4.5 cm on the ray from B.
4. From C, draw an arc of radius 5 cm (CD) to intersect the ray from A at point D.
5. Join C and D.
10. Construct a quadrilateral ABCD, such that AB = BC = CD = 6 cm, ∠B = 90° and ∠C = 120°
Steps of Construction:
1. Draw BC = 6 cm.
2. Construct ∠B = 90° and ∠C = 120°.
3. Cut AB = 6 cm on B-ray and CD = 6 cm on C-ray.
4. Join A and D.
11. Draw a parallelogram ABCD, with AB = 7 cm, AD = 6 cm and ∠DAB = 45°. Draw the perpendicular bisector of side AD and let it meet AD at point P. Also, draw the diagonals AC and BD, and let them intersect at point O. Join O and P. Measure OP.
Steps of Construction:
1. Construct parallelogram with AB=7, AD=6, ∠A=45°.
2. Construct perpendicular bisector of AD to find midpoint P.
3. Draw diagonals intersecting at O (midpoint of BD).
4. Join P and O.
Answer: OP = 3.5 cm.
Explanation: In △ABD, P is the midpoint of AD and O is the midpoint of BD. By the midpoint theorem, OP is parallel to AB and half of AB. OP = 7/2 = 3.5 cm.