Given that AB is a tangent of the circle with the center at X, AB = 12, and XD = 2.5, which is the length of DB?
Where is D ?
Where are A and B located on the tangent ?
Given that AB is a tangent of the circle with the center at X, AB = 12, and XD = 2.5, which is the length of DB?
To find the length of DB, we need to use the properties of tangents in a circle.
First, let's draw a diagram to better understand the problem:
D_____________B
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| / 12
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X
We know that AB is a tangent to the circle at point X, and XD is given as 2.5.
Now, let's apply the tangent theorem.
The tangent theorem states that if a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.
In this case, line AB is the tangent, and XD is the radius drawn to the point of tangency. Since the radius is perpendicular to the tangent, we know that angle DXB is a right angle (90 degrees).
Now, we can use the Pythagorean theorem to find the length of DB.
In right triangle DXB, we have one leg as XD = 2.5 and the hypotenuse as AB = 12. We want to find the length of the other leg, DB.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Applying the Pythagorean theorem:
DB^2 + XD^2 = AB^2
DB^2 + 2.5^2 = 12^2
DB^2 + 6.25 = 144
DB^2 = 144 - 6.25
DB^2 = 137.75
Taking the square root of both sides:
DB = √(137.75)
DB ≈ 11.73
Therefore, the length of DB is approximately 11.73 units.