Point B
lies in the exterior of circle A.
Angle ACB
and angle ADB
are right angles.
The diagram as described in the problem, where segment B C equals negative 2 x plus 85 and segment B D equals 5 x minus 48. What is the length of BC¯¯¯¯¯¯¯¯?
Enter the correct value.
To find the length of BC, we need to set the equations for BC and BD to be equal to 0. Since both angle ACB and angle ADB are right angles, we can use the Pythagorean theorem to solve for BC.
In triangle ABC:
BC^2 + AB^2 = AC^2
(2x - 85)^2 + (20)^2 = BC^2
4x^2 - 340x + 7225 + 400 = BC^2
4x^2 - 340x + 7625 = BC^2
In triangle ABD:
BD^2 + AB^2 = AD^2
(5x - 48)^2 + (20)^2 = BD^2
25x^2 - 240x + 576 + 400 = BD^2
25x^2 - 240x + 976 = BD^2
Since ACB and ADB are right angles, BC and BD are perpendicular to AB, making the triangles ACB and ADB right triangles.
BC^2 + BD^2 = CD^2
(4x^2 - 340x + 7625) + (25x^2 - 240x + 976) = CD^2
29x^2 - 580x + 8601 = CD^2
CD = sqrt(29x^2 - 580x + 8601)
Since point B lies in the exterior of circle A, BC and BD are diagonals of a cyclic quadrilateral, meaning that they are equal. Therefore, we can set BC equal to BD:
4x^2 - 340x + 7625 = 25x^2 - 240x + 976
21x^2 - 100x - 6649 = 0
Solving this quadratic equation gives x = 25.
Plugging this value of x back into the equation for BC, we get:
BC = 4(25)^2 - 340(25) + 7625
BC = 200 - 8500 + 7625
BC = 1325
Therefore, the length of BC¯¯¯¯¯¯¯¯ is 1325.