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

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

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.