X SAI TRIANGLE TEST 1

M.M. 30                                                                                                                                                     TIME:  75 MIN                                                                                                                                

1.        [3]   In the adjoining figure, P is a point on the side BC of AABC such that     MP ║ AB and NP ║ AC. If MN and CB produced meet in Q, prove that    PQ² = QB x QC.

2.        [3]   Using Basic Proportionality theorem  prove that the line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.

3.        [2]   In the adjoining figure, DE ║ AQ and DF ║ AR. Prove that EF ║ QR

.

4.        [3]   ABCD is a trapezium in which AB ║ DC and its diagonals intersect each other at O. Using Basic Proportionality theorem, prove that .

.

5.        [2]   Let ∆ABC ~ ∆DEF and their areas be 64 cm² and 121 cm² respectively. If EF = 13.2 cm , find BC.

6.        [3]   Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

7.        [3]   Prove that  in a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

8.        [3]   In the adjoining figure, ABD = CDB = PQB = 90°. If AB = x units, CD = y units and PQ = z units, prove

9.        [4]   In an equilateral triangle ABC, a point D is taken on base BC such that BD : DC = 2 : 1. Prove that 9 AD² = 7AB².                    

10.     [4]   In Figure given alongside, ABC is a triangle in which ∠ ABC < 90° and AD  BC. Prove that                    AC² = AB²  + BC²  – 2 BC . BD.

                 

OPTIONAL / HOME WORK

11.     In the adjoining figure, M is the mid-point of the side CD of parallelogram ABCD. The line BM is drawn intersecting AC in L and AD produced in E. Prove that    EL = 2 BL.

12.     Two isosceles triangles have equal vertical angles and their areas are in the ratio 16 : 25. Find the ratio of their corresponding>heights.

13.     Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding altitudes.