Final answer:
The sector ABD has the same area as the triangle ABC because the length of the arc BD is equal to the length of the tangent segment BC. The base and height of triangle ABC are equal to the base and radius of sector ABD, respectively.
Step-by-step explanation:
In the given question, we have an arc BD centered at point A and a tangent segment BC. We need to explain why the sector ABD has the same area as triangle ABC.
The area of a sector is given by the formula (1/2) * r^2 * theta, where r is the radius and theta is the central angle in radians. In this case, the radius of the sector ABD is represented by AB, and the central angle is the angle ABD.
On the other hand, the area of a triangle is given by the formula (1/2) * base * height. In triangle ABC, the base is BC and the height is the perpendicular distance from point A to line BC.
Since the chord BC is congruent to the arc BD, we can say that the base of the triangle BC is also congruent to the arc BD. Therefore, the base of the triangle and the arc have the same length.
Now, let's consider the height of the triangle. The height is the perpendicular distance from point A to line BC. Since the triangle is isosceles, this distance can be represented by the radius AB.
Thus, we can conclude that the base and height of triangle ABC are equal to the base and radius of sector ABD, respectively. Therefore, the area of the sector ABD is equal to the area of triangle ABC.