AB and AD are tangents of the circle with the center at C. The measure of BDC = 45o, and the circle has a diameter of 4. Which is the length of AB?
Tricky little question.
On my second attempt at the diagram I realized that in triangle BDC , the central angle had to be 90°
So the diagram really just consists of a square with sides 2.
So without any further calculations because of the properties of a square
AB = 2
The key lies in a proper diagram.
To find the length of AB, we need to use the properties of tangents and the given information.
Since AB and AD are tangents to the circle, they are perpendicular to the radius drawn from the center of the circle to the point of tangency. In this case, the radius is CO.
We know that the measure of angle BDC is 45°, which means that angle BCO is also 45° because angles in the same segment of a circle are equal. Since BCO is a right triangle, this means that angle BOC is 90°.
Now, we can use the properties of right triangles to find the length of AB. We have a right triangle BCO with angle BCO as 90° and sides BC and BO as the legs. We are given that the diameter of the circle is 4, so the radius CO is 4/2 = 2.
We can use the Pythagorean theorem to find the length of AB. The Pythagorean theorem states that the square of the hypotenuse (AB) is equal to the sum of the squares of the other two sides (BC and BO):
AB^2 = BC^2 + BO^2
From the triangle BCO, we can see that BC is equal to BO. Let's represent this length as x:
AB^2 = x^2 + x^2 = 2x^2
Since the diameter of the circle is 4 and the radius CO is 2, we have that BC = BO = 2. Substituting this value into the equation above:
AB^2 = 2^2 + 2^2 = 4 + 4 = 8
Taking the square root of both sides:
AB = √8 = √(4 * 2) = √4 * √2 = 2√2
Therefore, the length of AB is 2√2.