By using what we know about circles and distance between points, we will see that:
BD = 18 units
BC = 16.97 units
What is the distance from B to D?
Remember that the distance between two points (x₁, y₁) and (x₂, y₂) is given by:
[tex]d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
The distance from B to D, will be equal to the distance from B to A, minus the radius of the circle.
Because we know that the center of the circle is at (8, 10), and the edge of the circle is 4 units above the x-axis, we can assume that the closest point on the circle to the x-axis is:
(8, 4)
Then the radius of the circle is given by the distance between these two points:
[tex]r = \sqrt{(8 - 8)^2 + (10 - 4)^2} = 6[/tex]
Now we need to find the distance between A and B.
A = (8, 10)
B = (0, 10 - 16*√2)
The distance between these points is:
[tex]d = \sqrt{(8 - 0)^2 + (10 - 10 + 16*\sqrt{2})^2 } \\\\d = \sqrt{(8)^2 + (16*\sqrt{2})^2 } = 24[/tex]
And the distance from B to D is the distance from B to A, minus the radius of the circle, so we have:
BD = BA - r = 24 - 6 = 18
The distance from B to D is 18 units.
To get the value of BC, we use the fact that this is a cathetus of a right triangle with a hypotenuse of 18 units, and where the other side AC measures 6 units, then we have:
BC^2 + 6^2 = 18^2
BC = √(18^2 - 6^2) = 16.97 units.
If you want to learn more about circles, you can read:
https://brainly.com/question/1559324
