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SUPER-5 EDU ZONE AN INSTITUTE FOR YOUR WARDS PERFECTION BY: RAJESH KUMAR GIRI CLASSES FOR- XII/XI/X/IX [email protected] 100% RESULT FOR (B.COM/ENGINEERING) Website-www.rekhamadam.weebly.com Mobile no- 91-8527307169/91-9968042935 *MAIN FACTS* *Two figures having the same shape but not necessarily the same size are called similar figures. * All the congruent figures are similar but the converse is not true. * Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (i.e., proportion). * Two triangles are similar, if (i) Their corresponding angles are equal (ii) Their corresponding sides are in the same ratio (or proportion). * Basic Proportionality Theorem (B.P.T.) (Thales Theorem) In a triangle, a line drawn parallel to one side, to intersect the other sides in distinct points, divides the two sides in the same ratio. π¨π« π¨π¬ π¨π© π¨πͺ π¨π© π¨πͺ In ΞABC, if DE || BC then (i) = (ii) = (iii) = ππ ππ ππ ππ ππ ππ SPECIAL CLASSES FOR β MATHEMATICS (ENGINEERING/POLYTECHNIC/B.COM/B.CA/M.CA) CLASSES AVAILABLE FOR β ECONOMICS /ENGLISH/SCIENCE SUPER-5 EDU ZONE AN INSTITUTE FOR YOUR WARDS PERFECTION BY: RAJESH KUMAR GIRI CLASSES FOR- XII/XI/X/IX [email protected] 100% RESULT FOR (B.COM/ENGINEERING) Website-www.rekhamadam.weebly.com Mobile no- 91-8527307169/91-9968042935 Converse of Basic Proportionality Theorem If a line divides any two sides of a triangle in the same ratio, the line is parallel to the third side. In ΞPQR, if π·πΊ ππ = π·π» ππ , then ST || QR. * If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion). * If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion). * If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion). * If one angle of a triangle is equal to the one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion). * If a perpendicular is drawn from the vertex of the right angle of a right triangle to hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other. The ratios of the areas of two similar triangles are equal to the ratio of the squares of any two corresponding sides. * SPECIAL CLASSES FOR β MATHEMATICS (ENGINEERING/POLYTECHNIC/B.COM/B.CA/M.CA) CLASSES AVAILABLE FOR β ECONOMICS /ENGLISH/SCIENCE SUPER-5 EDU ZONE AN INSTITUTE FOR YOUR WARDS PERFECTION BY: RAJESH KUMAR GIRI CLASSES FOR- XII/XI/X/IX [email protected] 100% RESULT FOR (B.COM/ENGINEERING) Website-www.rekhamadam.weebly.com Mobile no- 91-8527307169/91-9968042935 * The areas of two similar triangles are in the ratio of the squares of the corresponding altitudes. * The areas of two similar triangles are in the ratio of the squares of the corresponding medians. * If the areas of two similar triangles are equal, then the triangles are congruent, i.e., equal and similar triangles are congruent. * The Pythagoras Theorem In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In figure, β B =90°, so, AC2 = AB2 + BC2. * Converse of the Pythagoras Theorem In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle. SPECIAL CLASSES FOR β MATHEMATICS (ENGINEERING/POLYTECHNIC/B.COM/B.CA/M.CA) CLASSES AVAILABLE FOR β ECONOMICS /ENGLISH/SCIENCE SUPER-5 EDU ZONE AN INSTITUTE FOR YOUR WARDS PERFECTION BY: RAJESH KUMAR GIRI CLASSES FOR- XII/XI/X/IX [email protected] 100% RESULT FOR (B.COM/ENGINEERING) Website-www.rekhamadam.weebly.com Mobile no- 91-8527307169/91-9968042935 MULTIPLE CHOICE QUESTIONS 1. In the figure, if π¨π¬ π¬π© = π¨π ππͺ , then we can conclude that (a) E and F are the mid-points of AB and AC respectively (b) EF || BC π¬π π¨π© (c) = π©πͺ π¨πͺ (d) None of the above 2. In the triangle ABC, DE || BC, then the length of DB is : (a) 2.5 cm (b) 5 cm (c) 3.5 cm (d) 3 cm SPECIAL CLASSES FOR β MATHEMATICS (ENGINEERING/POLYTECHNIC/B.COM/B.CA/M.CA) CLASSES AVAILABLE FOR β ECONOMICS /ENGLISH/SCIENCE SUPER-5 EDU ZONE AN INSTITUTE FOR YOUR WARDS PERFECTION BY: RAJESH KUMAR GIRI CLASSES FOR- XII/XI/X/IX [email protected] 100% RESULT FOR (B.COM/ENGINEERING) Website-www.rekhamadam.weebly.com Mobile no- 91-8527307169/91-9968042935 3. In ΞABC, if DE || BC, then the value of x is : (a) 4 (b) 6 (c) 8 (d) 9 4. In the trapezium ABCD, AB || CD, then the value of x is : (a) 2 (b) 3 (c) β2 (d) β3 5. In the given figure, PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm, then : (a) EF is not parallel to QR (b) EF || QR (c) Cannot say anything SPECIAL CLASSES FOR β MATHEMATICS (ENGINEERING/POLYTECHNIC/B.COM/B.CA/M.CA) CLASSES AVAILABLE FOR β ECONOMICS /ENGLISH/SCIENCE SUPER-5 EDU ZONE AN INSTITUTE FOR YOUR WARDS PERFECTION BY: RAJESH KUMAR GIRI CLASSES FOR- XII/XI/X/IX [email protected] 100% RESULT FOR (B.COM/ENGINEERING) Website-www.rekhamadam.weebly.com (d) None of the above Mobile no- 91-8527307169/91-9968042935 6. In the given figure, if ΞADE ~ ΞABC, then BC is equal to: (a) 4.5` (b) 3 (c) 3.6 (d) 2.4 7. In the given figure. ΞACB ~ ΞAPQ. If BC = 8 cm, PQ = 4 cm, BA = 6.5 cm and AP = 2.8 cm, then the length of AQ is: (a) 3.25 cm (b) 4 cm (c) 4.25 cm (d) 3 cm 8. If ΞABC ~ ΞPQR and β P = 50°, β B = 60°, then β R is : (a) 100° (b) 80° (c) 70° (d) cannot be determined 9.ΞABC ~ ΞDEF and the perimeters of ΞABC and ΞDEF are 30 cm and 18 cm respectively. If BC = 9 cm, then EF is equal to : (a) 6.3 cm (b) 5.4 cm (c) 7.2 cm (d) 4.5 cm 10.ΞABC ~ ΞDEF such that AB = 9.1 cm and DE = 6.5 cm. If the perimeter of ΞDEF is 25 cm, then perimeter of ΞABC is: (a) 35 cm (b) 28 cm (c) 42 cm (d) 40 cm 11. The areas of two similar triangles are 169 cm2 and 121 cm2, if the longest side of the larger triangle is 26 cm, then the longest side of the other triangle is : (a) 12 cm (b) 14 cm (c) 19 cm (d) 22 cm SPECIAL CLASSES FOR β MATHEMATICS (ENGINEERING/POLYTECHNIC/B.COM/B.CA/M.CA) CLASSES AVAILABLE FOR β ECONOMICS /ENGLISH/SCIENCE