# Learn Geomatory

Categories: Triangle

Constructions 157 21 Constructions 21. 1 INTRODUCTION One of the aims of the studying Geometry is to acquire the skill of drawing figures accurately. You have learnt how to construct geometrical figures namely triangles, squares and circles with the help of ruler and compasses. You have constructed angles of 30°, 60°, 90°, 120° and 45°. You have also drawn perpendicular bisector of a line segment and bisector of an angle. In this lesson we will extend our learning to construct some other important geometrical figures. 21. 2 OBJECTIVES After studying this lesson, the learner will be able to : divide a given line segment internally in a given ratio.

z Construct a triangle from the given data (i) SSS (ii) SAS (iii) ASA (iv) RHS (v) perimeter and base angles (vi) base, sum/difference of the other two sides and one base angle. (vii) two sides and a median corresponding to one of these sides. z Construct rectilinear figures such as parallelograms, rectangles, squares, rhombuses and trapeziums. z Construct a quadrilateral from the given data (i) four sides and a diagonal (ii) three sides and both diagonals (iii) two adjacent sides and three angles158 Mathematics (iv) three sides and two included angles v) four sides and an angle.

z Construct a triangle equal in area to a given quadrilateral. z Construct tangents to a circle from a point (i) outside it (ii) on it using the centre of the circle z Construct circumcircle of a triangle z Construct incircle of a triangle. 21. 3 EXPECTED BACKGROUND KNOWLEDGE We assume that the learner already knows how to use a pair of compasses and ruler to construct z angles of 30°, 45°, 60°, 90°, 105°, 120°.

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z the right bisector of a line segment z bisector of a given angle. z parallelograms, rhombuses, rectangles, and squares z a circle 21. DIVISION OF A LINE SEGMENT IN THE GIVEN RATIO INTERNALLY Construction 1 : To divide a line segment internally in a given ratio. Given a line segment AB. You are required to divide it internally in the ratio 2 : 3. We go through the following steps. Step 1 : Draw a ray AC making an acute angle with AB. Step 2 : Starting with A, mark off 5 points C1, C2, C3, C4 and C5 at equal distances from the point A. Step 3 : Join C5 and B. Step 4 : Through C2 (i. e. the second point), draw C2D parallel to C5B meeting AB in D. Fig. 21. 1 Then D is the required point which divides AB internally in the ratio 2 : 3 as shown in Fig. 1. 1Constructions 159 CHECK YOUR PROGRESS 21. 1 1. Draw a line segment 7 cm long. Divide it internally in the ratio 3 : 4. Measure each part. Also write the steps of construction. 2. Draw a line segment PQ = 8 cm. Find the point R on it such that PR = 3 4 PQ . [Hint : Divide the line segment PQ internally in the ratio 3 : 1]. 21. 5 CONSTRUCTION OF TRIANGLES Construction 2 : To construct a triangle when three sides are given (SSS) Suppose you are required to construct ? ABC in which AB = 6 cm, AC = 4. 8 cm and BC = 5 cm. We go through the following steps : Step 1 : Draw AB = 6 cm.

Step 2 : With A as centre and radius 4. 8 cm, draw an arc. Step 3 : With B as centre and radius 5 cm draw another arc intersecting the arc of Step 2 at C. Step 4 : Join AC and BC. Then ? ABC is the required triangle. [Note : You may take BC or AC as the base] Construction 3 : To construct a triangle, when two sides and the included angle is given (SAS) Suppose you are required to construct a triangle PQR in which PQ = 5. 6 cm, QR = 4. 5 cm and ? PQR = 60° For constructing the triangle, we go through the following steps. Step 1 : Draw PQ = 5. 6 cm Step 2 : At Q, construct an angle ? PQX = 60°

Step 3 : With Q as centre and radius 4. 5 cm draw an arc cutting QX at R. Step 4 : Join PR Then ? PQR is the required triangle. [Note : You may take QR = 4. 5 cm as the base instead of PQ] Construction 4. To construct a triangle when two angles and the included side are given (ASA). Let us construct a ? ABC in which ? B = 60°, ? C = 45° and BC = 4. 7 cm. Fig. 21. 2 Fig. 21. 3160 Mathematics To construct the triangle we go through the following steps : Step 1 : Draw BC = 4. 7 cm. Step 2 : At B, construct ? CBQ = 60° Step 3 : At C, construct ? BCR = 45° meeting BQ at A. Then ? ABC is the required triangle.

Note : To construct a triangle when two angles and any side (other than the included side) are given, we find the third angle (using angle sum property of the triangle) and then use above method for constructing the triangle. Construction 5 : To construct a right triangle, when its hypotenuse and a side are given. Let us construct a right triangle ABC, right angled at B, side BC = 3 cm and hypotenuse AC = 5 cm. To construct the triangle, we go through the following steps. Step 1 : Draw BC = 3 cm Step 2 : At B, construct ? CBP = 90°. Step 3 : With C as centre and radius 5 cm draw an arc cutting BP in A. Step 4 : Join AC ABC is the required triangle. Construction 6 : To construct a triangle when its perimeter and two base angles are given. Suppose we have to construct a triangle whose perimeter is 9. 5 cm and base angle are 60° and 45°. To construct triangle, we go through the following steps. Step 1 : Draw XY= 9. 5 cm Fig. 21. 6 Fig. 21. 5 Fig. 21. 4Constructions 161 Step 2 : At X, construct ? YXP = 30° [Which is 1/2 ? 60°] Step 3 : At Y, construct ? XYQ = 22? ° [Which is 1/2 ? 45°] Let XP and YQ intersect at A Step 4 : Draw right bisector of XA intersecting XY at B. Step 5 : Draw right bisector of YA intersecting XY at C.

Step 6 : Join AB and AC. ?ABC is the required triangle. Construction 7 : To construct a triangle when sum of two sides, third side and one of the angles on the third side are given. Suppose you are required to construct a triangle ABC. When AB + AC = 8. 2 cm, BC = 3. 6 cm and ? B = 45°. To construct the triangle, we go through the following steps : Step 1 : Draw BC = 3. 6 cm. Step 2 : At B, construct ? CBK = 45°. Fig. 21. 7 Step 3 : From BK, cut off BP = 8. 2 cm. Step 4 : Join CP. Step 5 : Draw right bisector of CP intersecting BP at A. Step 6 : Join AC ?ABC is required triangle. 162 Mathematics

Construction 8 : To construct a triangle when difference of two sides, the third side and one of the angles on the third side are given. Suppose we have to construct a ? ABC, in which BC = 4 cm ? B = 60°, AB – AC = 1. 2 cm. To construct the triangle we go through the following steps : Step 1 : Draw BC = 4 cm. Step 2 : Construct ? CBP = 60°. Step 3 : From BP cut off BK = 1. 2 cm. Step 4 : Join CK Step 5 : Draw right bisector of CK meeting BP produced at A. Step 6 : Join AC ?ABC is the required triangle. Construction 9 : To construct a triangle when its two sides and a median corresponding to one of these sides, are given.

Suppose you have to construct a ? ABC in which AB = 6 cm, BC = 4 cm and median CD = 3. 5 cm. We go through the following steps. Step 1 : Draw AB= 6 cm. Step 2 : Draw right bisector of AB meeting AB in D. Step 3 : With D as centre and radius 3. 5 cm draw an arc. Step 4 : With B as centre and radius 4 cm draw another arc intersecting the arc of Step 3 in C. Step 5 : Join AC and BC. Then ? ABC is the required triangle. CHECK YOUR PROGRESS 21. 2 1. Construct a ? DEF, given that DE = 5. 1 cm, EF = 4 cm and DF = 5. 6 cm. Write the steps of construction as well.

Note : You are also required to write the steps of construction in each of the remaining problems. 2. Construct a ? PQR, given that PR = 6. 5 cm, ? P = 120° and PQ = 5. 2 cm. Fig. 21. 8 Fig. 21. 9Constructions 163 3. Construct a ? ABC given that BC = 5. 5 cm, ? B = 75° and ? C = 45°. 4. Construct a right triangle in which one side is 3 cm and hypotenuse is 7. 5 cm. 5. Construct a right angled isosceles triangle in which one of equal side is 4. 8 cm. 6. Construct a ? ABC given that AB + BC + AC = 10 cm, ? B = 60°, ? C = 30°. 7. Construct a ? ABC in which AB = 5 cm, ? A = 60°, BC + AC = 9. 8 cm. 8.

Construct a ? LMN, when ? M = 30°, MN = 5 cm and LM – LN = 1. 5 cm. 9. Construct a triangle PQR in which PQ = 5 cm, QR = 4. 2 cm and median RS = 3. 8 cm. 21. 6 CONSTRUCTION OF RECTILINEAR FIGURES You are advised to draw rough sketch for the given data in each of the following constructions. You will observe, that it helps you to visualise/understand the steps of construction. Construction 10 : To construct a parallelogram when two adjacent sides and the included angle are given. Suppose that you have to construct a parallelogram in which the adjacent sides are 4 cm and 3 cm and included angle is 60°.

To construct the required parallelogram we go through the following steps : Step 1 : Draw AB = 4 cm Step 2 : At A, construct ? BAK = 60°. Step 3 : From AK cut off AD = 3 cm. Step 4 : With B and D as centres and radii equal to 3 cm and 4 cm respectively draw two arcs cutting each other at C. Step 5. Join CD and BC. Then ABCD is the required parallelogram. Construction 11 : To construct a rectangle when one of its diagonal and a side are given. Suppose that you have to construct a rectangle ABCD in which AB = 4 cm and AC = 5. 0 cm. Recall that in a rectangle, each angle is 90° and opposite sides are equal.

To construct the rectangle we go through the following steps. Step 1 : Draw AB = 4 cm. Fig. 21. 10164 Mathematics Step 2 : At B, draw ? ABK = 90°. Step 3 : With A as centre and radius 5 cm, draw an arc cutting BK at C. Step 4 : With C as centre and radius 4 cm, draw an arc. Step 5 : With A as centre and radius = BC, draw an arc cutting the arc drawn in Step 4 at D. Step 6 : Join DC and AD ABCD is the required rectangle. Construction 12 : To construct a square when its side is given. Suppose you have to construct a square PQRS in which PQ = 4. 4 cm. We have to follow the following steps to construct the square : Step 1 : Draw PQ = 4. cm. Step 2 : Construct ? PQT = 90° at Q. Step 3 : From QT cut off QR = 4. 4 cm. Step 4 : From P and R, draw two arcs of radii 4. 4 cm each to cut each other at S. Step 5 : Join PS and RS. PQRS is the required square. Construction 13 : To construct a parallelogram when two diagonals and the angle between them is given. Suppose that the lengths of two diagonals are 8 cm and 6 cm and the angle between them is 60°. Recall that diagonals of a parallelogram bisect each other. To construct the parallelogram, we go through the following steps : Step 1 : Draw AC = 8 cm. Step 2 : Draw right bisector of AC meeting it at O.

Step 3 : Construct ? COP = 60° and produce PO to Q. Step 4 : Cut off OB = OD = 3 cm (1/2 ? 6, length of second diagonal) from OP and OQ. Step 5 : Join AB, BC, AD and CD. ABCD is the required parallelogram. Fig. 21. 11 Fig. 21. 12 Fig. 21. 13Constructions 165 Construction 14 : To construct a rhombus when one diagonal and side are given. Suppose, you have to construct a rhombus, when one of its diagonal is 5. 5 cm and the side is 3. 3 cm. To construct the rhombus, we go through the following steps : Step 1 : Draw AC = 5. 5 cm. Step 2 : With A as centre and radius 3. 3 cm, draw two arcs one above AC and the other below AC.

Step 3 : With C as centre and radius 3. 3 cm draw two arcs one above AC and the other below AC intersecting the arcs of Step 2 in B and D respectively. Step 4 : Join AB, BC, CD and AD. ABCD is the required rhombus. Construction 15 : To construct a trapezium in which one of parallel sides, two non-parallel sides and the distance between parallel sides are given. Suppose you have to draw a trapezium in which one of parallel sides is 6 cm, two non-parallel sides are of length 4 cm and 5 cm and distance between parallel sides is 3 cm. To construct the trapezium we go through the following steps : Step 1 : Draw AB = 6 cm.

Step 2 : At A, draw AP? AB. Step 3 : From AP cut off AK = 3 cm. Step 4 : At K, draw KL? AK. Step 5 : With A and B as centres and radii 4 cm and 5 cm respectively draw two arcs cutting KL at D and C respectively. Step 6 : Join AD and BC. Then ABCD is the required trapezium. CHECK YOUR PROGRESS 21. 3 1. Construct a parallelogram if the lengths of its adjacent sides are 5. 5 cm and 4 cm and the included angle is 75°. Write steps of construction as well. Note : You are also required to write the steps of construction in each of the following problems. Fig. 21. 13 Fig. 21. 15166 Mathematics 2.

Step 2 : With A and B as centres and radii 3 cm and 5 cm respectively draw two arcs intersecting each other at D. Step 3 : Join AD. Step 4 : With B and A as centres and radii 2. 7 cm and 4. 8 cm draw two arcs intersecting each other at C. Step 5. Join DC and BC. ABCD is the required quadrilateral. Construction 18. To construct a quadrilateral when two adjacent sides and three angles are given. Suppose you have to construct a quadrilateral ABCD in which AB = 5 cm, BC = 4. 2 cm, ? B = 60°, ?C = 90° and ? A = 75°. Step 1 : Draw AB = 5 cm Step 2 : Construct ? BAP = 75°. Step 3 : Construct ? ABQ = 60°

Step 4 : Cut off BC = 4. 2 cm from BQ Step 5 : At C, construct ? BCR = 90° cutting AP at D. ABCD is the required quadrilateral. Construction 19 : To construct a quadrilateral when three sides and two included angles are given. Suppose you have to construct a quadrilateral PQRS in which PQ = 3 cm, QR = 4 cm, RS = 6 cm, ? Q = 120° and ? R = 90°. To construct the quadrilateral, we go through the following steps Step 1 : Draw QR = 4 cm. Step 2 : At Q and R, construct ? RQK = 120° and construct ? QRL = 90° Step 3 : From QK, cut off QP = 3 cm Fig. 21. 18 Fig. 21. 17 Fig. 21. 19168 Mathematics

Step 4 : From RL, cut off RS = 6 cm Step 5 : Join PS. Then PQRS is the required quadrilateral. Construction 20 : To construct a quadrilateral when four sides and an angle are given. Suppose that you have to construct a quadrilateral ABCD in which AB = 5. 5 cm, BC = 3. 5 cm, CD = 4 cm, AD = 5 cm, and ? A = 45°. To construct the quadrilateral ABCD, we go through the following steps. Step 1 : Draw AB = 5. 5 cm Step 2 : At A, construct ? BAK = 45° Step 3 : Cut off AD = 5 cm from AK. Step 4 : With B and D as centres and radii 3. 5 cm and 4 cm respectively, draw two arcs cutting each other at C.

Step 5 : Join BC and DC ABCD is the required quadrilateral. CHECK YOUR PROGRESS 21. 4 1. Construct a quadrilateral ABCD in which AB = 4 cm, BC = 5. 2 cm, CD = 5. 8 cm, DA = 6. 5 cm and AC = 7. 8 cm. Also write the steps of construction. Note : You are also required to write the steps of construction in each of the following problems. 2. Construct a quadrilateral ABCD in which AB = BC = CD = 4 cm, AC = 6 cm and BD = 7 cm. 3. Construct a quadrilateral ABCD in which AB = 6 cm, BC = 6. 5 cm, ? A = 45°, ? B = 120° and ? C = 90°. 4. Construct a quadrilateral PQRS in which PQ = QR = 5 cm, RS = 6 cm, ?

Q = 120° and ? R = 60° 5. Construct a quadrilateral ABCD in which AB = 4 cm, BC =3. 6 cm, CD = 5 cm, AD = 5. 2 cm and ? B = 45°. 6. Construct a quadrilateral PQRS in which PQ = 5 cm, RS = 6 cm, PS = 6. 2 cm, PR = 7 cm and the diagonal PR makes an angle of 30° with PQ. 21. 8 CONSTRUCTION OF A TRIANGLE EQUAL IN AREA TO A GIVEN QUADRILATERAL Construction 21 : To construct a triangle equal in area to a given quadrilateral. Suppose quad. ABCD is given Fig. 21. 20Constructions 169 We have to construct a triangle equal in area to the quadrilateral ABCD. For that we go through the following steps :

Step 1 : Join AC. Step 2 : Through D, draw a line segment DE || AC intersecting BC produced at E. Fig. 21. 21 Step 3 : Join AE. Then ? ABE is the required triangle. Construction 22 : To construct a quadrilateral and to construct a triangle equal in area to this quadrilateral. Suppose you have to construct a quadrilateral ABCD, in which AB = 3 cm, BC = 4. 2 cm, CD = 3. 6 cm, DA = 4. 5 cm and ? B = 135° and then to construct a triangle equal in area to this quadrilateral. We go through the following steps : Step 1 : Draw AB = 3 cm. Step 2 : Construct ? ABK = 135° and cut off BC = 4. cm from BK. Step 3 : With C and A as centres and radii 3. 6 cm and 4. 5 cm respectively draw two arcs intersecting each other at D. Step 4 : Join AD and CD. ABCD is the quadrilateral. Step 5 : Join DB. Step 6 : Through C draw CL || DB meeting AB produced in E. Step 7 : Join DE Then ? DAE is the required triangle. Fig. 21. 22170 Mathematics CHECK YOUR PROGRESS 21. 5 1. Draw a quadrilateral PQRS and construct a triangle equal in area to this quadrilateral. Also write the steps of construction. Note : You are required to write the steps of construction in each of the following problems. 2.

Construct a quadrilateral ABCD in which, AB = 4 cm, BC = 3. 5 cm, CD = 4. 2 cm, DA = 3 cm and AC = 6 cm. Construct a triangle equal in area to this quadrilateral. 3. Construct a rectangle ABCD in which AB = 5 cm and BC = 3. 5 cm. Construct a triangle, equal in area to the rectangle on AB as base. 21. 9 CONSTRUCTION OF TANGENTS TO A CIRCLE Construction 23 : To draw a tangent to a given circle at a given point on it using the centre of the circle. Suppose C be the given circle with centre O and a point P on it. You have to draw a tangent to the circle. We go through the following steps. Step 1 : Join OP.

Step 2 : At P, draw PT? OP. Step 3 : Produce TP to Q. Then TPQ is the required tangent. Construction 24 : To draw tangents to a circle from a given point outside it. Suppose C be the given circle and a point A outside it. You have to draw tangents to the circle from the point A. For that, we go through the following steps : Step 1 : Join OA. Step 2 : Draw the right bisector of OA. Let R be mid point of OA. Step 3 : With R as centre and radius equal to RO, draw a circle intersecting the given circle at P and Q. Step 4 : Join AP and AQ. Then AP and AQ are the required tangents. CHECK YOUR PROGRESS 21. 1. Draw a circle of 3 cm radius. Take a point A on the circle. At A, draw a tangent to the circle by using the centre of the circle. Also write steps of construction. 2. Draw a circle of radius 2. 5 cm. From a point P outside the circle, draw two tangents PQ and PR to the circle. Verify that lengths of PQ and PR are equal. Also write steps of construction. Fig. 21. 23 Fig. 21. 24Constructions 171 21. 10. CONSTRUCTION OF CIRCUMCIRCLE AND INCIRCLE OF A TRIANGLE. Construction 25 : To construct circumcircle of a triangle. Suppose a ? ABC is given. You have to draw a circumcircle of this triangle.

To construct it, we go through the following steps. Step 1 : Draw the given ? ABC. Step 2 : Draw right bisectors of any two sides ray BC and AC which meet each other at O. Step 3 : Join O with any one vertex say B. Step 4 : With O as centre and radius OB, draw a circle. This is the required circumcircle. Construction 26 : Construct a triangle with sides 4 cm, 5 cm and 6 cm. Draw a circumcircle of this triangle. We go through the following steps. Step 1 : BC = 5 cm. Step 2 : With B and C as centres and radii 4 and 6 cm respectively draw two arcs intersecting each other at A. Step 3 : Join AB and AC.

ABC is the required triangle. Step 4 : Draw right bisectors of AB and BC which meet each other at O. Step 5 : Join OB. Step 6 : With O as centre and radius OB, draw a circle. This is the required circumcircle. Construction 27 : To construct incircle of a triangle. Suppose you have to construct a triangle ABC with AB = 4 cm, BC = 3. 5 cm and ? B = 60° and draw its incircle. We go through the following steps Step 1 : Draw BC = 3. 5 cm Step 2 : Draw ? CBM = 60° Step 3 : From BM, cut off BA = 4 cm Fig. 21. 25 Fig. 21. 26 Fig. 21. 27172 Mathematics Step 4 : Join AC ABC is the required triangle. Step 5 : Draw bisectors of ? B and ?

C meeting each other at I. Step 6 : From I, draw IK perpendicular to BC meeting BC in D. Step 7 : With I as centre, and radius = ID, draw a circle. This is the required incircle. CHECK YOUR PROGRESS 21. 7 1. Construct a ? ABC with AB = 5 cm, BC = 4. 5 cm, ? B = 75° and draw its circumcircle. Also write steps of construction. Note : You are required to write the steps of construction in each of the following problems. 2. Construct an isosceles ? ABC with base BC = 4 cm and one of the equal sides AB = 3 cm and draw its circumcircle. 3. Construct a triangle with base 4 cm and base angles 60° and 75° and draw its incircle. . Construct an equilateral triangle of side 5 cm and draw its incircle. TERMINAL EXERCISE 1. Draw a line segment PQ = 8 cm long. Divide it internally in the ratio 3 : 5. Also write the steps of construction. Note : You are also required to write the steps of construction in each of the following problems. 2. Draw a line segment AB = 6 cm. Find a point C on AB such that AC : CB = 3 : 2. Measure AC and CB. 3. Construct a triangle with perimeter 14 cm and base angle 60° and 90°. 4. Construct a right angled triangle whose hypotenuse is 8 cm and one if its other two sides is 5. 5 cm. 5. Construct a ?

ABC in which BC = 3. 5 cm, AB + AC = 8 cm and ? B = 60°. 6. Construct a ? ABC in which AB = 4 cm, ? A = 45° and AC – BC = 1 cm. 7. Construct a parallelogram having its diagonals as 5 cm and 6 cm and angle between them is 45°. 8. Construct a rectangle with sides 8 cm and 6 cm. Measure the length of its two diagonals. 9. Construct a square with one side 4. 2 cm. Measure its diagonals. Constructions 173 10. Construct a square if its diagonal is 7 cm. 11. Construct a rhombus when the diagonals are 9 cm and 7 cm. 12. Construct a trapezium with one of parallel sides as 6 cm, two non-parallel sides as 4 cm and 4. cm and distance between parallel sides as 2. 5 cm. 13. Construct a trapezium ABCD in which AB = 8 cm, BC = 4. 5 cm, CD = 4 cm, ? B = 60° and AB || CD. [Hint : ? C = 180° – 60° = 120°] 14. Draw a quadrilateral ABCD in which AB = 4 cm, BC = 5 cm, CD = 4. 5 cm, DA = 5 cm and AC = 7 cm. Construct a triangle equal in area to this quadrilateral. 15. Draw a circle of diameter 6 cm. From a point P outside the circle, draw two tangents to the circle. 16. Construct triangle with sides 4 cm, 3 cm and 5. 5 cm and draw it circumcircle. 17. Construct a right angled triangle with base 4 cm and hypotenuse 6 cm and draw its incircle.