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

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**Q.1Â Â Â Â ABC is a triangle. Locate a point in the interior of ABC which is equidistant from all the vertices of ABC.**

**Sol.**

Let OD and OE be the perpendicular bisectors of sides BC and CA of 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 ofÂ ABC.

**Q.2Â Â Â Â Â 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 and respectively intersecting AC and AB at E and F respectively.

Since O lies on BE, the bisector of , hence O will be equidistant from AB and BC. Again O lies on the bisector CF of .

Hence , O will be equidistant from BC and AC. Thus, O will be equidistant from AB, BC and CA.

Â **Q.3Â Â Â Â In a huge park, people are concentrated at three points (see figure) : **

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **A : where there are different slides and swings for children, **

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **B : near which a man- made lake is situated. **

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **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.

Â **Q.4Â Â Â Â 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 figureÂ (ii)Â number of such triangle is 300.

Figure (ii) has more triangles.