Q2 of 18 Page 163

An isosceles triangle with its vertices on a circle is shown in this picture.


What is the perimeter of the circle?

The figure is shown below:



Let OA = radius of circle.


AD = 1 cm


OA = r


∴ OD = 1-r


OA = OB = r


In triangle OBD,


∠ODB = 90°


AB = AC and AD is perpendicular on BC from point A.


∴ BD = DC =


∴ By pythagoras theorem,


(Hypotenuse)2 = (One side)2 + (Other side)2


∴ (OB)2 = (OD)2 + (BD)2


∴ r2 = (1-r)2 + ()2


∴ r2- (1-r)2 =


∴ (r + {1-r})(r-{1-r}) = {a2-b2 = (a + b) × (a-b)}


∴ 1 × (2r-1) =


∴ 2r =


∴ r =


If r = radius of circle


Circumference =


∴ Perimeter =


More from this chapter

All 18 →
3

The perimeter of a circle of diameter 2 metres was measured and found to be about 6.28 metres. How do we compute the perimeter of a circle of diameter 3 metres, without measuring?

 1

In the picture below, a regular hexagon, square and a rectangle are drawn with their vertices on a circle. Calculate the perimeter of each circle.

 3

In all the pictures, below, centres of the circles are on the same line. In the first two pictures, the small circles are of the same diameter.


Prove that in all pictures, the perimeters of the large circle is the sum of the perimeters of the small circles.

 4

In this picture, the circles have the same centre and the line drawn is a diameter of the large circle. How much more is the perimeter of the large circle than the perimeter of the small circle?