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Triangles : Exercise 7.5 (Optional) (Mathematics NCERT Class 9th)


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Q.1     ABC is a triangle. Locate a point in the interior of \Delta ABC which is equidistant from all the vertices of \Delta ABC.
Sol.

Let OD and OE be the perpendicular bisectors of sides BC and CA of \Delta ABC.
Therefore O is equidistant from two ends B and C of line- segment BC as O lies on the perpendicular bisector of BC. Similarly, O is equidistant from C and A.

35Thus , the point of intersection O of the perpendicular bisectors of sides BC, CA and AB is the required point which is equidistant from vertices A,B,C of  \Delta ABC.


Q.2      In a triangle locate a point in its interior which is equidistant from all the sides of the triangle.
Sol.

Let BE and CF be the bisectors of \angle ABC and \angle ACB respectively intersecting AC and AB at E and F respectively.
34Since O lies on BE, the bisector of \angle ABC, hence O will be equidistant from AB and BC. Again O lies on the bisector CF of \angle ACB.
Hence , O will be equidistant from BC and AC. Thus, O will be equidistant from AB, BC and CA.


 Q.3     In a huge park, people are concentrated at three points (see figure) :
                   A : where there are different slides and swings for children,
                   B : near which a man- made lake is situated.
                   C : which is near to a large parking and exit.

36Where should an icecream parlour be set up so that maximum number of persons can approach it?
Sol.

37The parlour should be equidistant from A, B and C, for which the point of intersection of perpendicular bisector should be located.
Thus O is the required point which is equidistant from A, B and C.


 Q.4     Complete the hexagonal and star shaped Rangolies [see figure (i) and (ii)] by filling them with as many equilateral triangles of side 1 cm as you can. Count the number of triangles in each case. Which has more triangles?
38Sol.

On filling each figure with equilateral triangles of side 1 cm, we find in figure-
(i) number of such triangles is 150, and in figure  (ii)  number of such triangle is 300.

Figure (ii) has more triangles.
39



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