**Triangles**

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

**Solution:**

Let us consider a ∆ABC.

Draw l, the perpendicular bisector of AB.

Draw m, the perpendicular bisector of BC.

Let the two perpendicular bisectors l and m meet at O.

O is the required point which is equidistant from A, B and C.

Note: If we draw a circle with centre O and radius OB or OC, then it will pass through A, B and C. The point O is called circumcentre of the triangle.

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

**Solution:**

Let us consider a ∆ABC.

Draw m, the bisector of ∠C.

Let the two bisectors l and m meet at O.

Thus, O is the required point which is equidistant from the sides of ∆ABC.

Note: If we draw OM ⊥ BC and draw a circle with O as centre and OM as radius, then the circle will touch the sides of the triangle. Point O is called incentre of the triangle.

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

**A:**where these 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 exist.

Where should an ice-cream parlor be set? up so that maximum number of persons can approach it?

[**Hint **The parlour should be equidistant from A, B and C.]

**Solution:**Let us join A and B, and draw l, the perpendicular bisector of AB.

Now, join B and C, and draw m, the perpendicular bisector of BC. Let the perpendicular bisectors l and m meet at O.

The point O is the required point where the ice cream parlour be set up.

Note: If we join A and C and draw the perpendicular bisector, then it will also meet (or pass through) the point O.

**Question 4.Complete the hexagonal and star shaped Rangolies [see Fig. (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?**

**Solution:**It is an activity.

We require 150 equilateral triangles of side 1 cm in the Fig. (i) and 300 equilateral triangles in the Fig. (ii).

∴ The Fig. (ii) has more triangles.