1. Each interior angle of a polygon is 135. How many sides does it have? askiitians
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1 Class: VIII Subject: Mathematics Topic: Practical Geometry No. of Questions: Each interior angle of a polygon is 135. How many sides does it have? (A) 10 (B) 8 (C) 6 (D) 5 (B) Interior angle =. 135 =. 135n = 180n = n n = 8 It has 8 sides 2. The bisectors of two adjacent angles of a parallelogram intersect at? (A) 30 (B) 45 (C) 60 (D) 90 (D) We know that the opposite sides and the angles in a parallelogram are equal. Also Its adjacent sides are supplementary i.e, sum of the sides is equal to 180 Now, the bisectors of these angles from a triangle, whose two angles are Page 1
2 and or = (90 ) Sum of angle of triangle is = 180 o = = 90 Hence the bisector intersect at right angles 3. In square PQRS, if PQ = (2x +3 )cm and QR = (3x -5) cm then? (A) x = 4 (B) x = 5 (C) x = 6 (D) x = 8 (D) All sides of square are equal PQ = QR (2x + 3) = (3x 5) 2x +3x = -5 3 x = 8cm 4. The angles of quadrilateral are in the ratio 1 : 3 : 7 : 9. The measure of the largest angle is? (A) 63 (B) 72 (C) 81 (D) None of these (D) Page 2
3 Let the angles be (x), (3x), (7x) and (9x) Sum of the angles of the quadrilateral is 360 x + 3x +7x +9x = x = 360 x = 18 Angles are : 3x =3 18 = 54 7x = 7 18 =126 9x = 9 18 = How many diagonals are there in a hexagon? (A) 6 (B) 8 (C) 9 (D) 10 (C) Hexagon has six sides Number of diagonals = = (where n number of sides) = 9 Page 3
4 6. Construct a quadrilateral PQRS in which PQ = 4.5 cm, PQR = 120,QR =3.8 cm QRS = 100 and QRS = 60 Steps of construction 1. Draw PQ = 4.5cm 2. Make PQX = 120 using compass. 3. With Q as centre and radius 3.8cm, draw an arc, cutting Q at R. 4. Make QRY = 100 using protractor 5. Make QPZ = 60 so that PZ and RY intersect each other at the point S. Then,PQRS is the required quadrilateral, drawn alongside. 7. Construct a rhombus with side 4.2cm and one of its angles equal to 65. X A D Y B C 4.2 cm Clearly, adjacent angle =( ). Step of construction. (I) Draw BC=4.2 cm (II) Make CBX =115 and BCY =65. (III) Set off BA = 4.2 cm along BX and CD =4.2 cm along CY. Page 4
5 (IV) Join AD> Then, ABC is the required rhombus 8. Construct a square ABCD, each of whose diagonal is 5.2 cm. X D Y Steps of construction A O C B (I) Draw AC = 5.2cm (II) Draw the right bisector XY of AC, meeting AC at O. (III) (IV) From O set off OB = (5.2) cm = 2.6 cm along OY and OD = 2.6 cm along OX Join AB, BC, CD and DA. Then,ABCD is the required square. 9. Construct a rectangle ABCD in which side BC = 5 cm and diagonal BD = 6.2cm. Steps of construction (I) Draw BC = 5cm. Page 5
6 (II) Draw CX BC (III) With B as centre and radius 6.2 cm draw an arc, cutting CX at D. (IV) Join BD (V) With D as centre and radius 5cm, draw an arc (VI) With B a centre and radius equal to CD draw another arc, cutting the previous arc at A. (VII) Join AB and AD. Then ABCD is the required rectangle. 10. Construct a parallelogram whose diagonal are 5.4 cm and 6.2cm and an angle between them is 70. Steps of Construction: (i) Draw AC = 5.4 cm. (ii) Bisect AC at O. (iii) Make COX = 70 and produce XO to Y. (iv) Set off OB = 1/2 (6.2) = 3.1 cm and OD = 1/2 (6.2) =3.1 cm as shown. (v) Join AB, BC, CD and DA. Page 6
7 11. Construct a quadrilateral ABCD in which AB= 4 cm, BC =3.8 cm, AD= 3cm, diagonal AC = 5 cm and diagonal BD = 4.6 cm Steps of Construction (I) (II) (III) (IV) (V) (VI) (VII) Draw AB = 4 cm With A as center and radius equal to 5 cm, draw an arc. With B as center and radius equal to 3.8 cm, draw another arc, cutting the previous arc at C Join BC With A as center and radius equal to 3 cm, draw an arc With B as center and radius equal to 4.6 cm draw another arc, cutting the previous arc at D. Join AD and CD. 12. Is it possible to construct a quadrilateral ABCD in which AB= 3cm, BC = 4cm, CD = 5.5 cm, DA = 6cm and BD = 9cm? If not, give reason. D 5.5cm 6cm 9cm C A 3cm B 4cm The measurements must be such that the sum of any two sides of a triangle is greater than the third side. Page 7
8 AB= 3cm, BD = 9cm, and DA = 6cm AB + AD = 3cm + 6 cm = 9 cm= BD ABD cannot be constructed the quadrilateral ABCD cannot constructed 13. Construct a trapezium ABCD in which AB= 6 cm, BC = 4cm, CD = 3.2 cm, B = 75 and DC AB Steps of construction (I) Draw AB = 6cm (II) Make ABX = 75 (III) With B as centre, draw an arc at 4cm.Name that point as C. (IV) AB CD ABX + BCY = 180 BCY = Make BCY = 105 At C, draw an arc of length 3.2cm (V) Join A and D Thus, ABCD is the required trapezium Page 8
9 14. Construct a parallelogram,one of whose sides is 4.4 cm and whose diagonal are 5.6cm and 7 cm. Measure the other side. Step of construction : (I) Draw AB= 4.4 cm (II) With A as centre and radius 2.8cm, draw an arc (III) With B as the centre and radius 3.5cm, draw another arc, cutting the previous arc at point O. (IV) Join OA and OB (V) Produce OA to C, such that OC = OA. Produce OB to D, such that OB=OD (VI) Join AD, BC and CD Thus.ABCD is the required parallelogram.th e other side is 4.5 cm in length. 15. In each of the figures given below, ABCD is a rhombus. Find the values of x and y in each case. (i) ABCD is a rhombus and AC is its diagonal In ABC, AB = BC BAC = BCA [sides of a rhombus] (Angles opposite to equal sides) Page 9
10 BAC = x But BAC + BCA + ABC = 180 (Sum of angles of a triangle) x + x = 180 2x= = 62 x = = 31 Diagonal AC bisects A and C x=y y = 31. Hence x= 31, y= 31 (II) In rhombus ABCD Diagonal bisect each other at right angles AOB = 90 Now in AOB, AOB + OAB + OBA = x= x= 180 x= = 48 In ABD, AB = AD ABD = ADB x= y = 48 Hence x= 48, y= Construct a rhombus the length of whose diagonal are 6cm and 8cm. Page 10
11 Step of construction (I) Draw AC = 6cm (II) Draw a perpendicular bisector (XY) of AC, which bisects AC at O. (III) OB= (8)cm OB = 4cm And OD = (8)cm (IV) OD = 4cm Draw an arc of length 4cm on OX and name that point as B. Draw an arc of length 4cm on OY and name that point as D. Join AB, BC, CD and AD. Thus, ABCD is the required rhombus. 17. Construct a parallelogram, one of whose sides is 5.2 cm and whose diagonals are 6 cm and 6.4 cm. Page 11
12 Steps of Construction: (i) Draw AB = 5.2 cm. (ii) With A as center and radius 3.2 cm, draw an arc. (iii) With B as center and radius 3 cm draw another arc, cutting the previous arc at O. (iv) Join OA and OB. (v) Produce AO to C such that OC = AO and produce BO to D such that OD = OB. (vi) Join AD, BC and CD. Then, ABCD is the required parallelogram. 18. Construct a quadrilateral ABCD in which AB = 3.6 cm, ABC = 80, BC = 4 cm, BAD = 120 and AD = 5 cm. Steps of Construction: (I) Draw AB = 3.6 cm. (II) Make ABX = 80 using protractor. (III) With B as center and radius equal to 4 cm, draw an arc, cutting BX at C. (IV) Make BAY = 120 using protractor/compass. (V) With A as center and 5 cm as radius, draw an arc, cutting AY at D. Join CD. Then, ABCD is the required quadrilateral. 19. Construct a parallelogram ABCD in which AB = 6 cm, BC = 4.5 cm and diagonal AC = 6.8 cm. Page 12
13 Steps of Construction: (i) Draw AB = 6 cm. (ii) With A as center and radius 6.8 cm, draw an arc. (iii) With B as center and radius 4.5 cm draw another arc, cutting the previous arc at C. (iv) Join BC and AC. (v) With A as center and radius 4.5 cm, draw an arc. (vi) With C as center and radius 6 cm draw another arc, cutting the previously drawn arc at D. (vii) Join DA and DC. Then, ABCD is the required parallelogram. Page 13
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