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**Question 1. ABC is a triangle. Locate a point in the interior of Î”ABC which is equidistant from all the vertices of Î”ABC. Soultion:** Let us consider a Î”ABC.

Draw â€˜lâ€™ the perpendicular bisector of AB.

Draw â€˜mâ€™ the perpendicular bisector of BC.

Let the two perpendicular bisectors â€˜lâ€™ and â€˜mâ€™ meet at O. â€˜Oâ€™ is the required point which is equidistant from A, B and C.

Note: If we draw a circle with centre â€˜Oâ€™ and radius OB or OC, then it will pass through A, B and C.

**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 Î”ABC.

Draw â€˜lâ€™ the bisector of âˆ B.

Draw â€˜mâ€™ the bisector of âˆ C.

Let the two bisectors l and m meet at O. Thus, â€˜Oâ€™ is the required point which is equidistant from the sides of Î”ABC.

required point which is equidistant from the sides of Î”ABC.**Note: **If we draw OM âŠ¥ BC and draw a circle with O as centre and OM as radius, then the circle will touch the sides of the triangle.

**Question 3. In a huge park, people are concentrated at three points (see figure): A : where there are different slides and swings form children,**

**B : near which a man-made lake is situated, C: which is near to a large parking and exit. Where should an ice cream parlour 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 â€˜lâ€™ the perpendicular bisector of AB.

Now, join B and C, and draw â€˜mâ€™ the perpendicular bisector of BC. Let the perpendicular bisectors â€˜lâ€™ and â€˜mâ€™ meet at â€˜Oâ€™. The point â€˜Oâ€™ is the required point where the ice cream parlour be set up.

Note: If we join â€˜Aâ€™ and â€˜Câ€™, and draw the perpendicular bisectors, 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 get the 150 equilateral triangles in the figure

(i) and 300 equilateral triangles in the figure

(ii). âˆ´ The figure

(ii) has more triangles.

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