Page 1 :
GEOMETRY AND ALGEBRA, , , , IMPORTANT POINTS, , , , 1., , ABCD is a parallelogram., AB is parallel to CD, , D c, AD is parallel to BC. [_/, , But, AB and CD are not parallel to x axis., AD and BC are not prallel to y axis., , Knowing the coordinates of three vertices we can, find the coordinates of the fourth vertex., , The shift of x coordinates of A and B is equal to the, shift of x coordinates of C and D. Shift of y coordi, nates of A and B is equal to the shift of y coordinates, of CandD A, , R, the coordiantes of its vertices., Join P,QandR c, Now we can see the sucieoan PROB., , Knowing P, R and Q we can write the coordinates of B., Similarly the coordinates of Aand C., , Knowing the mid points of the, sides of a triangle we can find, , A point P(x, y) divides the line joining two points A, (x, y,) and B(x, a in the ratio nN, , ay: Ya), m, y=y; te ae, , A A(x, Ys), , m, X= X, +—(% m+n, , If P(x, y) is the mid point of the line joining A (x,, y,), , and B(x,, y,) then,, /\, D Cc, , ae Yea, 2° 2, , Median of a triangle is the line, joining a vertex to the mid point, , of the opposite side., , There is a point on a median, , which divides it in the ratio 2:1, , Gis the centroid of the triangle 8, , , , 10., , If a line makes an angle x, with x axis, tan x will be its, slope., , If (x,, ¥,), (x, y,) are the, points on a line., , Ys, -%, , , , , , _Y2, Slope = x,, , The slopes of prallel lines are equal., , The concept of slope can be used to check whether, some points are in a line or not A (x,, y,), B(x, y.), and C(x,, y,) if slope of the line AB is equal to the, slope of line BC, the points A, B and C are on a line., y-coordinates of all points on x axis are zero., , y = 0 is considered as the equation of x axis., , All points on a line parallel to x axis are equal., Example (3, 4), (2, 4), (0, 4) ... are the points on the, line parallel to x axis., , y = 4s the equation of this line., , Similarly, x = 0 is the equation of y axis, , x= kis the equation of the line parallel to y axis., The linear relationship between the coordinates of, , points on aline is considered as the equation of this, line., , Example:, , (1, 1), (2, 2), (3, 3) are the points on a line., The equation of the line is y = x, , The equation of a line can be determined by using, the concept of slope., (1, 2) and (3, 4) are two, , points on a line. (34), , P(x, y), Let (x, y) be a point on it., , Slope of AP = Slope of AB A(t, 2), , (y-2)x=x-1, y-2-x+1=0, y-x-1=0, y=x+1, This is the equation of the line.

Page 2 :
11. Let (0, 0) lie the centre, of a circle P(x, y) lie a point (& y), on this circle. = Hh, The distance between (0, 0) and (x, y) will be the, tadius of the circle., (x- 0? + (y-0P =r, P= x? + y?, , 12. If (a, b) is the centre and (x, y), is a point on a circle,, (x-a)? + (y-b)? =P? is the, equation of the circle., , , , , , , , , , Find out the co-ordinates of the fourth vertex of the, parallelogram given in this figure., , , , (1,3) 4, Co-ordinates of fourth vertex is (6, 6), Qn. 2, The figure shows a parallelogram with the coordinates of its vertices:, Prove that x, + x, =x, +X, andy, + ¥, =Y2 + Ys, , 5 ¥), , , , (r1,¥1), , Xs The side AB is parallel to CD. The shift of x coordi, nates of A and B is equal A (x,, y,) to the shift of x, , coordinates of C and D., , xX, = x, = X i x,, , % 7 X, r x # X, , The shift of y coordinates, , Yo Yq = Ya > Ya, , Y, + ¥3 = Yo * Ya, , , , Alem), , Qn.3, Find out the coordinates of the fourth vertex of the, , parallelogram in which the lines obtained by joining, the points (x,, y,), (x, y,) and the origin are adjacent, sides. :, , , , , , , (%) %s Ys Ye), iY:, “(Ki Pe Ys), , (%., 0), , , , oO, (0, 0), The co-ordinate of fourth vertex = (x,+X,, y, +y,), , Qn. 4 ., Prove that the sum of a squares of all the sides in, , any parallelogram is equal to the sum of squares of, the diagonals., , XD, , (x2.92)B. c, (% + %¥i FY), , 0 A, 0) (x), OA? + OB? = x? +y? +x3 +3, OA = BC, OB =AC, , AB? = (Xp =x) +(Y2 -y,), , OC? = (Xp -X4)? +(Y2 Ya)", , OB? +002 =2Axf +x5 +7 +97), , OA? + OB? + AC? +BC? = AB? +0C?, , Qn. 5 S, The smaller triangle in the figure is formed by joining, , the mid points of the sides of the bigger triangle., Find out the co-ordinates of all the vertex of the bigger, , triangle. =, , (3,3) D c (54), , E, A (4,2), , NWABCD is a prallelogram., A(x, y) is the fourth vertex., X=4+3-5=2, y=3+2-4=4, A (2, 1) is the vertex, E (x, y) is a vertex, BDCE is a parallelogram, X=4+5-3=6, y=2+4-3=3

Page 3 :
E (6, 3), , F(x, y) is a vertex, , DBCF is a parallelogram, x=3+5-4=4, y=3+4-2=5, , F (4, 5), , , , , , ACTIVITIES IN THE TEXT - Page 217, , , , , , , , Qn. 1, , Acircle is drawn with the line joining (2, 3) and (6, 5), as diameter. What are the coordinates of the centre, of the circle?, , XD Let O (x, y) be the centre of the circle, , 2, (6. 5), , (2,3), , Gn. 2 ., , The coordinates of two opposite vertices of a parallelogram are (4, 5) and (1, 3). What are the coordinates of the point of intersection of its diagonals?, , Xw The diagonals of the parallelogram bisect each other., , Mid point of AC is the point of intersection of the, diagonals., , P(x y)=P|, , , , Qn. 3, , The coordinates of the vertices of a quadrilateral,, , taken in order, are (2, 1), (5, 3), (8, 7), (4,9)., , a) Find the coordinates of the midpoints of all four, sides., , b) Prove that the quadrilateral got by joining these, midpoints is a parallelogram., , XB a) Let P (x, y) be the mid point of AB., , 2 BHe p18, x= FO, , P(x, y) =P (3. 2], , , , , , Al2, 1), , Q (x, y) is the mid point of BC, , awy=a (52.337) «n(2), , R(x, y) = RG, 8), S (x, y) is the mid point of AD, , 2+4 1+9 6 10, Sq MEST? PS 2 2, , = S(3, 5), b) InPQRS, , (2. 2}, ofS. 5}, R(6, 8), 8 (3,5), , PR and QS are the diagonals., Mid point of PR is (x, y), , c, , ot8 248, , y=] 2° 2, , {2, 4, Mid point of QS is (x, y), 13, ot? 646, , (wy)=] 2 ' 2, , 29, 4, Digonals bisect each other, Therefore PQRS is a, parallelogram., , Qn. 4, , In the figure, the midpoints of the large quadrilateral, are joined to form the smaller quadrilateral within:, , i) Find the coordinates of the fourth vertex of the, smaller quadrilateral.

Page 4 :
ii) Find the coordinates of the other three vertices, ofthe larger quadrilateral., , , , (0), , Nw Consider D (x, y), , x+2, 3= 736 =x 42,x=4, , 1, aS eqy+ty=5, D(,5), , , , A(2,1), Consider C (x, y), , 4 ¥, S56, 44+x=12, x=8, , 544, At ne, 5 +y=12,y=7, , C(x y)=C (8, 7), Consider B (x, y), , , , g= 8%, g4x=18, x= 10, , 2, 7+y, , ma, B(x, y) = B(10, 3), , 2+10 1+3, P (x, y)=P (= 5), , 7+y = 10, y=3, , 2, , =°( 2.4) = P6, 2), , Qn. 5, The coordinates of the vertices of a triangle are, , (3,5), (9, 13), (10,6). Prove that this triangle is isosceles. Calculate its area., , Xp A(3, 5), B(9, 13), C(10, 6), , AB= (9-3)? + (13-5)? = Je? +8?, = 436+ 64, , , , peat, BC = (10-9) + (6-13), , = featP, = +49, = /60, AC = (0-3)? + (6-5), =F, = /49+1, = 50, BC =AC, -. A ABC is isosceles, : 34+9 54+13, Mide point of ABis [3-9, C(10, 6), -(# %), V2 2) aes, es, = (6, 9), (), AG,5) 10 Be, 13), CD = (6-10)? + (9-6), = (ayer?, = 25, , =5, , Area =4%10%5, , 25, , Qn.6, The centre of a circle is (1, 2) and a point on itis (3,, 2). Find the coordinates of the other end of the diameter through this point., , 3+x 34+x=2, 18, XW = 3 a,, 3+x, 25 4a34+y y=, , <r (-), A(3, 2) BY)

Page 5 :
Qn.1, , The co-ordinates of the point A and B are (3, 2), (8,7), on the line AB., , i) Find out the co-ordinates of the point P such that, AP: PB=2:3, , li) Find out the co-ordinates of point Q such that, , AQ: QB=3:2, , , , an m, NBD x=% +5, , +n, , (x, -%), B(8,7), 2 Boge PO y),, 3+ ons (8-3) 5, , 2, , HS+exb=S A(3, 2), , m, m+n, , , , y =¥y+———(y, -%), , 2, =2+=(7+72), , =2+2x5=4, P(x, y) = P (5, 4), , ‘ m, il) x=x, +o Oe -%), B(8,7), , 3, =3+ -(83456 3), =3+3=6 3, , | y 22+3(7=2) AG, 2), =2#3=5, Q& y)=Q6,5), , Qn. 2, , Find the coordinates of the points which divide the, line joining (1,6) and (5, 2) into three equal parts., XW P(x,, y,) divides AB in the ratio 1:2, , 1, x, = 5+3 (1-8), , (1,6), , % =$+ix(-4), 3 QX, Ya), , 4, , 3, , =5, PU, ¥,), , w]e, , 1 A(5.2), y, = 2 +,(6-2), , 1 4 10, n2tax4 #2to= 3, , , , 411 10, ny) Fg? a, Q (x, y,) divides the line Ab in the ratio 2 :1, , x,=8+3(1-5) 25+ 2%-4, , 26, elo, wlNn, , 2 2, yy 82+5(8-2)=2+5%4, , ols, , 8, sive a, *3, , 7 14, 2(%y Yo) = off, 4), Qn. 3, , The coordinates of the vertices of a triangle aré, (1, 5), (3, 7), (1, 1). Find the coordinates of its, , , , centroid., Ny D(x, y) is the mid point of AB, , _-1+3) 547, er Oe, x=1, y=6, D(x, y) = D (1, 8), , ACTS) Dixy) Ba), D(1,6), , Let G (x, y) be the centroid., , It divides the median in the ratio 2:1, 2, , xate gat, , 2 2 13, =1+£(6-1)=14+2x5=—, y tal ) +g%8 3, , 13, G(x, y) =G (s 2), , Qn. 4, , Calculate the coordinates of the point P in the, picture:, , (0,3), , , , (0,0) (4,0), XU AABP, ACBP isa, , PA=a, PC =6