Total time:3hrs darsha vidyalay hunashyal P. M.M=80 STION- 10 1=10 1) Name the point in a triangle that touches all sides of given triangle. Write its symbol of representation. 2) Where is thocenter of a right angled triangle ( ) right angled at located? 3) In above figure is a cyclic quadrilateral. If 0 100 and 0 70, find. 100 0 70 0 4) Prove that angle subtended by semicircle is 90 o. 5) If and F are to and respectively. Then find if =10cm, is 4cm and F is 5cm. F 6) Prove that area of triangle is half area of parallelogram if between two parallel lines and having common base. 7) Find area of a triangle with sides 3cm, 4cm, 5cm using heron s formula. Page 1
8) Name the type of triangle where orthocenter, in-center, and circumcenter are collinear. 9) Prove that opposite sides of parallelogram are equal. 10) How many solutions are possible when two lines are parallel? Section (2 mark) 11) erive is a triangle in which is midpoint of and is the midpoint of. Prove that area of 1 area of. 4 12) 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. Show that a median of a triangle divides it into two triangles of equal area. 13) Prove that the opposite angles of cyclic Quadrilateral are supplement of each other. 14) iagonals and of a trapezium with intersect each other at O. Prove that ar (O) = ar (O). Page 2
and are points on sides and respectively of triangle such that ar () = ar (). Prove that. 15) Parallelogram and rectangle F are on the same base and have equal areas. Show that the perimeter of the parallelogram is greater than that of the rectangle. Section (3 mark) 16) In the bellow figure divides in the ratio of 1:3 and =. etermine the value of x. 108 0 17) In figure bellow m and n are two plane mirrors perpendicular to each other. Show that the incident ray is parallel to the reflected ray. m n Page 3
18) Prove that perimeter of the triangle is greater than the sum of its three medians. Prove that Perimeter of the triangle is greater than the sum of the three altitudes. 19) If two circles intersect in two points, prove that the line joining the centres is the perpendicular bisector of the common chord. 20) Of any two chords of a circle show that the one which is larger, is nearer to the centre. Of any two chords of a circle show that the one which is nearer to the centre, is larger. 21) If, &F are midpoints of sides,, respectively then prove that F is a parallelogram. F O is the centre of the circle of the circle of radius 5cm, OP,, =6cm, =8cm and chords on opposite side. etermine PQ. 22) Prove that: a) qual chord subtend equal angle at centre. b) qual chords are equidistant from centre. Page 4
23) In triangle, is the mid-point of.pis any point of. Q P meets in Q.Show that M. Q P Section (6 mark) 24) If, F, G, H are respectively, the mid-points of sides,, and of parallelogram. Show that the quadrilateral FGH is a) a parallelogram b) its area is half area of. 25) is a parallelogram and O is any point in its interior. Prove that: a) ar( O) ar ( O ) ar ( O ) ar ( O ) b) ar( F ) ar ( ) In Fig. 9.34, is a right triangle right angled at., FG and MN are squares on the sides, and respectively. Line segment X. meets at Y. Show that: a) ar (YX) = 2 ar (F) b) M c) ar () = ar (MN) + ar (FG) Page 5
26) isectors of angle, and of a triangle intersect its circumcircle at, and F respectively. Prove that angles of triangle F are and 0 0 90, 90 2 2 0 90. 2 27) Prove that: (2+2+2) a) If trapezium has two non-parallel sides equal then it is cyclic. b) For two triangles having same base, their areas are proportional to their heights drawn from the vertex opposite to the common base.() c) rea of equilateral triangle is 4 3 2 a when length of one side is a. (4+2) a) is a parallelogram. and F are midpoints of sides and respectively. F and intersect at P;F and intersect at Q. Prove that i) F is a parallelogram. ii) PQF is a parallelogram. b) Prove parallelogram has opposite sides and angles equal. 28.In Fig bellow, XY, X and Y. Prove that: ar( X ) ar( Y ). 29. In parallelogram, and F are two points on side and respectively. Show that ar( F ) ar ( ). Page 6