# PRACTICE QUESTIONS CLASS IX: CHAPTER 4 LINEAR EQUATION IN TWO VARIABLES

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1 PRACTICE QUESTIONS CLASS IX: CHAPTER 4 LINEAR EQUATION IN TWO VARIABLES 1. Find the value of k, if x =, y = 1 is a solution of the equation x + 3y = k.. Find the points where the graph of the equation 3x + 4y = 1 cuts the x-axis and the y-axis. 3. At what point does the graph of the linear equation x + y = 5 meet a line which is parallel to the y-axis, at a distance units from the origin and in the positive direction of x-axis. 4. Determine the point on the graph of the equation x + 5y = 0 whose x-coordinate is 5 times its ordinate. 5. Draw the graph of the equation represented by the straight line which is parallel to the x-axis and is 4 units above it. 6. Draw the graphs of linear equations y = x and y = x on the same cartesian plane. What do you observe? 7. Determine the point on the graph of the linear equation x + 5y = 19, whose ordinate is 1 1 times its abscissa. 8. Draw the graph of the equation represented by a straight line which is parallel to the x-axis and at a distance 3 units below it. 9. Draw the graph of the linear equation whose solutions are represented by the points having the sum of the coordinates as 10 units. 10. Write the linear equation such that each point on its graph has an ordinate 3 times its abscissa. 11. If the point (3, 4) lies on the graph of 3y = ax + 7, then find the value of a. 1. How many solution(s) of the equation x + 1 = x 3 are there on the : (i) Number line (ii) Cartesian plane 13. Find the solution of the linear equation x + y = 8 which represents a point on (i) x-axis (ii) y-axis 14. For what value of c, the linear equation x + cy = 8 has equal values of x and y for its solution. 15. The following values of x and y are thought to satisfy a linear equation : x 1 y 1 3 Draw the graph, using the values of x, y as given in the above table. At what point the graph of the linear equation (i) cuts the x-axis. (ii) cuts the y-axis. 16. The Autorikshaw fare in a city is charged Rs 10 for the first kilometer Rs 4 per kilometer for subsequent distance covered. Write the linear equation to express the above statement. Draw the graph of the linear equation. 0. The work done by a body on application of a constant force is the product of the constant force and the distance travelled by the body in the direction of force. Express this in the form of a linear equation in two variables and draw its graph by taking the constant force as 3 units. What

2 is the work done when the distance travelled is units. Verify it by plotting the graph. 1. The following values of x and y are thought to satisfy a linear equation, Write the linear equation. x 6 6 y 6 6 Draw the graph, using the values of x, y as givenin the above table. At what point the graph of the linear equation (i) cuts the x-axis. (ii) cuts the y-axis.. Draw the graph of the linear equation 3x + 4y = 6. At what points, the graph cuts the x-axis and the y-axis. 3. The force exerted to pull a cart is directly proportional to the acceleration produced in the body. Express the statement as a linear equation of two variables and draw the graph of the same by taking the constant mass equal to 6 kg. Read from the graph, the force required when the acceleration produced is (i) 5 m/sec, (ii) 6 m/sec. 4. If the temperature of a liquid can be measured in Kelvin units as x K or in Fahrenheit units as y F, the relation between the two systems of measurement of temperature is given by the linear equation y 9 ( x 73) 3 5 (i) Find the temperature of the liquid in Fahrenheit if the temperature of the liquid is 313 K. (ii) If the temperature is 158 F, then find the temperature in Kelvin. 5. The linear equation that converts Fahrenheit (F) to Celsius (C) is given by the relation C 5 F (i) If the temperature is 86 F, what is the temperature in Celsius? (ii) If the temperature is 35 C, what is the temperature in Fahrenheit? (iii) If the temperature is 0 C what is the temperature in Fahrenheit and if the temperature is 0 F, what is the temperature in Celsius? (iv) What is the numerical value of the temperature which is same in both the scales? 6. Draw the graph of x + y = 7 and x y = on the same graph. 7. If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a. 8. The taxi fare in a city is as follows: For the first kilometre, the fare is Rs 8 and for the subsequent distance it is Rs 5 per km. Taking the distance covered as x km and total fare as Rs y, write a linear equation for this information, and draw its graph.

3 9. Solve the equation x + 1 = x 3, and represent the solution(s) on (i) the number line, (ii) the Cartesian plane. 30. Give the geometric representations of y = 3 as an equation (i) in one variable (ii) in two variables 31. Give the geometric representations of x + 9 = 0 as an equation (i) in one variable (ii) in two variables 3. The force applied on a body is directly proportional to the acceleration produced in the body. Write an equation to express this situation and plot the graph of the equation. 33. Draw the graphs of the equations x y + 1 = 0 and 3x + y 1 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region. 34. Draw the graphs of the equations y = x and y = x in the same graph paper. Find the coordinates of the point where two lines intersect.

4 PRACTICE QUESTIONS CLASS IX: CHAPTER 8 QUADRILATERALS 1. In the below figure, bisectors of B and D of quadrilateral ABCD meets CD and AB, produced at P and Q respectively. Prove that P + Q = 1 ( ABC + ADC). In ABC, AD is the median through A and E is the midpoint of AD. BE produced meets AC in F such that BF DK. Prove that AF = 1 AC 3 3. In a parallelogram, the bisectors of any two consecutive angles intersects at right angle. Prove it. 4. In a quadrilateral ABCD, AO and BO are the bisectors of A and B respectively. Prove that AOB = 1 ( C + D) 5. ABCD is a square E, F, G, H are points on AB, BC, CD and DA respectively such that AE = BF = CG = DH. Prove that EFGH is a square. 6. ABCD is a parallelogram. If its diagonals are equal, then find the value of ABC.

5 7. In the below figure, ABCD is a parallelogram and DAB = If the bisector AP and BP of angles A and B respectively meet P on CD. Prove that P is the midpoint of CD. 8. In the below given figure, ABCD is a parallelogram and E is the midpoint of side BC, DE and AB when produced meet at F. Prove that AF = AB. 9. ABC is right angle at B and P is the midpoint of AC and Q is any point on AB. Prove that (i) PQ AB (ii) Q is the midpoint of AB (iii) PA = 1 AC 10. The diagonals of a parallelogram ABCD intersect at O. A line through O intersects AB at X and DC at Y. Prove that OX = OY. 11. ABCD is a parallelogram. AB is produced to E so that BE = AB. Prove that ED bisects BC. 1. If ABCD is a quadrilateral in which AB CD and AD = BC, prove that A = B. 13. Diagonals AC and BD of a parallelogram ABCD intersect each other at O. If OA = 3 cm and OD = cm, determine the lengths of AC and BD. 14. In quadrilateral ABCD, A + D = What special name can be given to this quadrilateral? 15. All the angles of a quadrilateral are equal. What special name is given to this quadrilateral? 16. In ABC, AB = 5 cm, BC = 8 cm and CA = 7 cm. If D and E are respectively the mid-points of AB and BC, determine the length of DE. 17. Diagonals of a quadrilateral ABCD bisect each other. If A = 35 0, determine B.

6 18. Opposite angles of a quadrilateral ABCD are equal. If AB = 4 cm, determine CD. 19. In the below figure, it is given that BDEF and FDCE are parallelograms. Can you say that BD = CD? Why or why not? 0. In the above right sided figure, ABCD and AEFG are two parallelograms. If C = 55 0, determine F. 1. Angles of a quadrilateral are in the ratio 3 : 4 : 4 : 7. Find all the angles of the quadrilateral.. In the below figure, X and Y are respectively the mid-points of the opposite sides AD and BC of a parallelogram ABCD. Also, BX and DY intersect AC at P and Q, respectively. Show that AP = PQ = QC. 3. One angle of a quadrilateral is of 108 and the remaining three angles are equal. Find each of the three equal angles. 4. ABCD is a trapezium in which AB DC and A = B = Find angles C and D of the trapezium. 5. The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is. 60 Find the angles of the parallelogram. 6. ABCD is a rhombus in which altitude from D to side AB bisects AB. Find the angles of the rhombus. 7. E and F are points on diagonal AC of a parallelogram ABCD such that AE = CF. Show that BFDE is a parallelogram. 8. ABCD is a parallelogram and DAB = If the bisectors AP and BP of angles A and B respectively, meet at P on CD, prove that P is the midpoint of CD. 9. ABCD is a parallelogram. AM and BN are respectively, the perpendiculars from A and B to DC and CD produced. Prove that AM = BN.

7 30. ABCD is a parallelogram. L and M are points on AB and DC respectively and AL = CM. Prove that LM and BD bisect each other. 31. Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD such that AP = CQ (see below figure). Show that AC and PQ bisect each other. 3. In the below figure, P is the mid-point of side BC of a parallelogram ABCD such that BAP = DAP. Prove that AD = CD. 33. D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle ABC. Show that DEF is also an equilateral triangle. 34. E is the mid-point of the side AD of the trapezium ABCD with AB DC. A line through E drawn parallel to AB intersect BC at F. Show that F is the mid-point of BC. 35. PQ and RS are two equal and parallel line-segments. Any point M not lying on PQ or RS is joined to Q and S and lines through P parallel to QM and through R parallel to SM meet at N. Prove that line segments MN and PQ are equal and parallel to each other. 36. Prove that If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. 37. Prove that A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel. 38. Prove that A quadrilateral is a parallelogram if its opposite angles are equal. 39. Show that the diagonals of a rhombus are perpendicular to each other. 40. Two parallel lines l and m are intersected by a transversal p. Show that the quadrilateral formed by the bisectors of interior angles is a rectangle. 41. Show that the bisectors of angles of a parallelogram form a rectangle. 4. If the diagonals of a parallelogram are equal, then show that it is a rectangle. 43. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

8 44. Show that the diagonals of a square are equal and bisect each other at right angles. 45. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. 46. In the adjoining figure, ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD. If AQ intersects DP at S and BQ intersects CP at R, show that: (i) APCQ is a parallelogram. (ii) DPBQ is a parallelogram. (iii) PSQR is a parallelogram. 47. The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral. 48. Prove that The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it. 49. Prove that The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side. 50. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. 51. ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle. 5. ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that (i) D is the mid-point of AC (ii) MD AC (iii) CM = MA = 1 AB 53. In ABC, D, E and F are respectively the mid-points of sides AB, BC and CA. Show that ABC is divided into four congruent triangles by joining D, E and F. 54. Prove that the quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.

9 55. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that: APCQ is a parallelogram

10 PRACTICE QUESTIONS CLASS IX: CHAPTER 9 AREAS OF gms AND TRIANGLES 1. ABCD is a parallelogram x and y are midpoints of BC and CD respectively. Prove that ar( AXY) = 3 ( gm ABCD) 8. The medians BE and CF of a triangle ABC intersect at G. Prove that ar( GBC) = area of quadrilateral AFGE. 3. In fig. PQRS is a square and T and U are respectively, the midpoints of PS and QR. Find the area of OTS if PQ = 8cm. 4. In fig. ABCD, ABFE and CDEF are parallelograms. Prove that ar( ADE) = ar( BCF). 5. In fig. ABCD is a trapezium in which AB CD. Prove that area of AOD = area of BOC.

11 6. The diagonals of parallelogram ABCD intersect at a point O. Through O, a line is drawn to intersect AD at P and BC at Q. Show PQ divides the parallelogram into two parts of equal area. 7. In the fig. O is any point on the diagonal BD of the parallelogram ABCD. Prove that ar( OAB) = ar( OBC). 8. Show that the diagonals of a parallelogram divide it into four triangles of equal area. 10. Prove that Two parallelograms on the same base and between the same parallels are equal in area. 11. Prove that Two triangles on the same base and between the same parallels are equal in area. 1. Prove that a median of a triangle divides it into two equal parts 13. If a triangle and a parallelogram are on the same base and between the same parallels, then prove that the area of the triangle is equal to half the area of the parallelogram. 14. If E,F,G and H are respectively the mid-points of the sides of a parallelogram ABCD, show that ar (EFGH) = 1 ar (ABCD) 15. Show that the diagonals of a parallelogram divide it into four triangles of equal area. 16. D and E are points on sides AB and AC respectively of ABC such that ar (DBC) = ar (EBC). Prove that DE BC. 17. XY is a line parallel to side BC of a triangle ABC. If BE AC and CF AB meet XY at E and F respectively, show that ar (ABE) = ar (ACF) 18. In fig. P is a point in the interior of a parallelogram ABCD. Show that (i) ar (APB) + ar (PCD) = 1 ar (ABCD) (ii) ar (APD) + ar (PBC) = ar (APB) + ar (PCD) 38. In Fig. ABCD is a quadrilateral and BE AC and also BE meets DC produced at E. Show that area of ADE is equal to the area of the quadrilateral ABCD.

12 39. Diagonals AC and BD of a trapezium ABCD with AB DC intersect each other at O. Prove that ar (AOD) = ar (BOC). 40. Diagonals AC and BD of a quadrilateral ABCD intersect each other at O such that ar (AOD) = ar (BOC). Prove that ABCD is a trapezium. 41. ABCD is a trapezium with AB DC. A line parallel to AC intersects AB at X and BC at Y. Prove that ar (ADX) = ar (ACY). 4. In the above Fig. AP BQ CR. Prove that ar (AQC) = ar (PBR). 43. Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar (AOD) = ar (BOC). Prove that ABCD is a trapezium. 44. The medians BE and CF of a triangle ABC intersect at G. Prove that the area of GBC = area of the quadrilateral AFGE. 45. Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that ar (APB) ar (CPD) = ar (APD) ar (BPC). 46. ABCD is a trapezium in which AB DC, DC = 30 cm and AB = 50 cm. If X and Y are, respectively the mid-points of AD and BC, prove that ar (DCYX) = 7 ar (XYBA) In ABC, if L and M are the points on AB and AC, respectively such that LM BC. Prove that ar (LOB) = ar (MOC) 48. If the medians of a ABC intersect at G, show that ar (AGB) = ar (AGC) = ar (BGC) = 1 ar (ABC) Prove that the area of rhombus is equal to half the rectangle contained by its diagonals. 50. A point O inside a rectangle ABCD is joined to the vertices. Prove that the sum of the areas of a pair of opposite triangles so formed is equal to the sum of the other pair of triangles.

13 51. The medians BE and CF of a triangle ABC intersect at G. Prove that area of GBC = area of quadrilateral AFGE. 5. A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented. 53. P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and R is the mid-point of AP, show that (i) ar (PRQ) = 1 ar (ARC) (ii) ar (RQC) = 3 ar (ABC) (iii) ar (PBQ) = ar (ARC) A quadrilateral ABCD is such that the diagonal BD divides its area in two equal parts. Prove that BD bisects AC. 55. D, E and F are respectively the mid-points of the sides BC, CA and AB of a ABC. Show that (i) BDEF is a parallelogram. (ii) ar (DEF) = 1 ar (ABC) (iii) ar (BDEF) = 1 ar (ABC). 4

14 PRACTICE QUESTIONS CHAPTER 10: CIRCLES 1. Prove that Equal chords of a circle subtend equal angles at the centre.. Prove that Chords of a circle which subtends equal angles at the centre are equal. 3. Prove that The perpendicular from the centre of a circle to a chord bisects the chord. 4. Prove that The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. 5. Prove that Chords equidistant from the centre of a circle are equal in length 6. Prove that Chords of a circle which are equidistant from the centre are equal 7. Prove that Of any two chords of a circle then the one which is larger is nearer to the centre. 8. Prove that Of any two chords of a circle then the one which is nearer to the centre is larger. 9. Prove that line joining the midpoints of two equal chords of circle subtends equal angles with the chord. 10. Prove that if two chords of a circle bisect each other they must be diameters. 11. If two chords of a circle are equally inclined to the diameter through their point of intersection, prove that the chords are equal. 1. Prove that The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. 13. Prove that Angles in the same segment of a circle are equal. 14. Prove that Angle in a semicircle is a right angle. 15. Prove that Arc of a circle subtending a right angle at any point of the circle in its alternate segment is a semicircle. 16. Prove that Any angle subtended by a minor arc in the alternate segment is acute and any angle subtended by a major arc in the alternate segment is obtuse. 17. Prove that If a line segment joining two points subtends equal angles at two other points lying on the same side of the line segment, the four points are concyclic. 18. Prove that Circle drawn on any one side of the equal sides of an isosceles trainlge as diameter bisects the side. 19. Prove that The sum of either pair of opposite angles of a cyclic quadrilateral is 180º. 0. Prove that If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.

15 1. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle.. Prove that If two sides of a cyclic quadrilateral are parallel, then the remaining two sides are equal and the diagonals are also equal. 3. Prove that If two opposite sides of cyclic quadrilateral are equal, then the other two sides are parallel. 4. Prove that If two non parallel sides of a trapezium are equal, it is cyclic. 5. Prove that The sum of the angles in the four segments exterior to a cyclic quadrilateral is equal to 6 right angles. 6. Two circles with centres A and B intersect at C and D. Prove that ACB = ADB. 7. Bisector AD of AC of ABC passes through the centre of the circumcircle of ABC. Prove that AB = AC. 8. In the below figure A, B and C are three points on a circle such that angles subtended by the chords AB and AC at the centre O are 80 0 and 10 0 respectively. Determine BAC. 9. In the above right-sided figure, P is the centre of the circle. Prove that XPZ = ( XZY + YXZ). 30. Prove that the midpoint of the hypotenuse of a right triangle is equidistant from its vertices. 31. In the below figure ABCD is a cyclic quadrilateral, O is the centre of the circle. If BOD = 160 0, find BPD. 3. Prove that in a triangle if the bisector of any angle and the perpendicular bisector of its opposite side intersect, they will intersect on the circumcircle of the triangle.

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