A convex quadrilateral is inscribed in a circle so that one of its sides is a diameter of the cirlce. The length of the two sides of the quadrialteral which have exactly one endpoint on this diameter are 7 and 20. If the length of the longest side of this quadrilateral is 25, find the length of the fourth side of the quadrilateral.
+128474 See the following image :
We have a circle with the equation x^2 + y^2 = 12.5^2
And
AB = 25 BC = 7
Then triangle ABC is a 7 - 24 - 25 right triangle
The area of this triangle = 7 * 24 / 2 = 84 units^2
And we can find the height of this triangle as
84 = (1/2)(25) h
84 / 12.5 = h = 168/25 = 6.72 units = the y value of point C
So....the x value of point C is given as x = - sqrt [12.5^2 - 6.72^2 ] = -10.54
Andl let AD = 20
Then triangle ADB will also be a right triangle with sides 15 - 20 - 25
And the area of this triangle = (1/2)(15)(20) = 150
And we can find the height of this triangle as
150 = (1/2) 25 * h
150/12.5 = h = 12 = the y coordinate of point D
So the x coordinate of D is given as x = - sqrt [ 12.5^2 - 12^2] = -3.5
So.....the length of the missing side = chord CD =
sqrt [ (-10.54 + 3.5)^2 + ( 12 - 6.72)^2 ] = 8.8 units
