The line segment XZ has a length of 16 units. Hence, the 3rd option is the correct choice. Computed using the tangent to the circle and the Pythagoras Theorem.
How is a tangent inclined to the circle?
A tangent to a circle is always perpendicular to it, that is, the radius drawn from the center of the circle to the tangent, is always perpendicular to it.
What is the Pythagoras Theorem?
According to the Pythagoras Theorem, in a right triangle, the square of the hypotenuse, that is, the side opposite to the right angle, is equal to the sum of the squares of the legs, that is, the other two sides.
How to solve the question?
In the question, we are asked to find the length of the line segment XZ.
Firstly, we name the center of the circle O.
The diameter of the circle is given to be 12 units.
Thus, its radius = 12/2 = 6 units,
Joining the radius OW, we get a right triangle OWZ, as the radius from the center of the circle to the tangent is always perpendicular.
In the right triangle OWZ, by Pythagoras Theorem, we can write:
OZ² = OW² + WZ² {Since, OZ is the hypotenuse},
or, (k + 6)² = 6² + (k + 4)² {Since, OZ = OY + YZ = Radius + k = 6 + k, and OW = Radius = 6},
or, k² + 12k + 36 = 36 + k² + 8k + 16 {Using the formula (a + b)² = a² + 2ab + b²},
or, k² + 12k - k² - 8k = 36 + 16 - 36 {Rearranging},
or, 4k = 16 {Simplifying},
or, k = 4 {Simplifying}.
Now, XZ = XY + YZ = Diameter + k = 12 + 4 = 16 units.
Thus, line segment XZ has a length of 16 units. Hence, the 3rd option is the correct choice. Computed using the tangent to the circle and the Pythagoras Theorem.
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