1. In a triangle ABC altitude from C to AB is CF= 8 units and AB has length 6 units. If M and P are midpoints of AF and BC. Find the length of PM.

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1 1. In a triangle ABC altitude from C to AB is CF= 8 units and AB has length 6 units. If M and P are midpoints of AF and BC. Find the length of PM. 2. Let ABCD be a cyclic quadrilateral inscribed in a circle of radius R. If BC=2AB, CD=2AD and ADC=60 if ratio of area of quadrilateral 4 3 with area of circle be then find k? k 3. let C, C, C bethree circles such that ( C, C ),( C, C ),( C, C ) pair of circles touch each other externaly at point s P, Q, R respectively. Line joining PQ and PR meets C at M, N. Pr ovethat MN is a diameter of circlec. 3 3

2 4. Let S and S be 2 concentric circles and ABC, A B C be any 2 equilateral triangles inscribed in them respectively. If P and P are any two points on S and S respectively, show that P A + P B + P C = A P + B P + C P. 5. Suppose ABC is an acute angled triangle with b>c, BE is altitude through B and AD is bisector of angle A. prove that angle CED>45?. 6. PQRis anisosceles triangle whoseequal sides PQ and PR areat right angles. S and t are point son PQ suchthat QS 6SP and QT 2 TP. Pr ovethat 2 PRT PRS PRQ. 7. Square PQRS is inscribed into triangle ABC so that vertices P and Q lie on sides AB and AC and vertices R and S lie on BC. Express the length of the square s side through a and h the altitude through A.

3 8. A rectangular sheet of paper ABCD is folded along line segment EF so that C ends up with midpoint Of side AB. If AB=240 and BC=288, determine the length of EF. 9. Let O, I be the cicumcentre and incentre of ABC. AI is drawn, and extended to intersect the larger circle in D. Then prove that CD=ID=BD. 10. Points D, E, F are taken on sides AB, BC, CA of triangle ABC so that AD:DB=BE:CE=CF:FA=1:n. Prove that ratio of the area of triangle 2 n n 1 DEF to ABC is equal to 2 (n+1). 11. In a triangle ABC a=c+1 and b=c-1. Prove that line joining incentre and centroid is parallel to AB. Also find the length of IG. 12. Let AN be any line drawn through A in a Triangle ABC. Let BM, CN perpendiculars are drawn from B and C to AN. If D be mid-point of BC, prove that MD=ND.

4 13. In a right angled isosceles triangle ABC A (angle B=90degree), a point M is taken in such a way that Angle MAB=angle MBA=15degree. Find angle CMB. 14. In ABC, AC > BC, CM is the median, and CH is the altitude emanating from C, Determine the measure of MCH if ACM and BCH each have measure P 1 2 B A 4 3 M C H C B let ABCD be a parallelogram, Also a ABF is made on base AB. The perpendicular FE is drawn from F to CD such that FE=7.5 units and E lies interior of CD if BC=5 units, AF=6.5units, find the minimum length of AB? 16. Let P be an interior point of a triangle ABC. Extend AP, BP, CP to meet BC, CA,AB respectively in D, E, F. Suppose the areas of the triangles APE, APF and BPD are equal, then we can establish that P is the centroid of triangle ABC

5 A 17. In a triangle ABC, let incircle of it touches sides BC, AB, AC at D, E, F respectively. Prove that lines AD, BE, CF are concurrent. F E. 18. Let OAB is a right angled triangle ( AOB =90 ). B D Midpoint of OA is M. P is a point on circumcircle of the traingle and QP OB where Q lies on line segment OB. If BMQ QMO find BPQ? C 19. PQRS is suchthat a circle canbeinscribed touching all the sides of the. Incircleof PQR, SPR touches side PR at X, Y find XY 20. We have locate a point inside triangle ABC from which the sum of perpendicular distances to sides are maximum.

6 21. From a point M inside an equilateral triangle ABC perpendiculars MP, MQ, MR are dropped to sides BC, CA, AB respectively. Then AR BP CQ AQ BR CP AR BP CQ AQ BR CP 22. ABC is an isoscles triangle in which AB=AC. BD is the length of median through B. BD=m given and area of triangle ABC is maximum. 23. If sides a, b, c of a triangle are in A.P and bisecto c a B meets side AC at E then AE= and CE= In a square ABCD, there are any points P, Q, R,S on sides AB,BC,CD,AD respectively. If AP=BQ=CR=DS. Prove that quad.pqrs is a square. 25. PROVE THAT IF A TRIANGLE IS INSCRIBED IN A CIRCLE PRODUCT OF TWO SIDES=PRODUCT OF DIAMETER AND PERPENDICULAR DISTANCE OF 3 SIDE FROM VERTEX.

VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER)

BY:Prof. RAHUL MISHRA Class :- X QNo. VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER) CIRCLES Subject :- Maths General Instructions Questions M:9999907099,9818932244 1 In the adjoining figures, PQ