A point,X ,is 25 feet from the center of a circle.If the diameter of the circle is 14 feet,what is the length (in feet)of a tangent from point X to the circle?
a. square root 29
b. 18
c. 15 square root 2
d. 24
e. square root 673
please answer and explain
To find the length of the tangent from point X to the circle, we can utilize the property that a tangent line is perpendicular to the radius that intersects the point of tangency.
First, let's draw a diagram to visualize the situation. On the diagram, mark point X, the center of the circle, and the diameter of 14 feet.
Since the diameter is 14 feet, we can find the radius (r) by dividing the diameter by 2. In this case, r = 14/2 = 7 feet.
Now, let's consider the right triangle formed by the radius, the line segment connecting the center of the circle to point X, and the tangent line. Using the Pythagorean theorem, we can find the length of the tangent line (T).
The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.
In this case, the radius (r) is one of the legs of the triangle (perpendicular to the tangent line), and the line segment connecting the center of the circle to point X is the other leg.
Applying the Pythagorean theorem:
r^2 + X^2 = T^2
7^2 + 25^2 = T^2
49 + 625 = T^2
674 = T^2
To find T, we take the square root of both sides:
T = √674 ≈ 25.96 feet
Now, we can look at the given answer choices and determine which one is closest to the calculated length of the tangent line.
a. √29 ≈ 5.39 feet
b. 18 feet
c. 15√2 ≈ 21.21 feet
d. 24 feet
e. √673 ≈ 25.94 feet
From the given answer choices, option (e) √673 is the closest to the calculated length of the tangent line, which is approximately 25.96 feet.
Therefore, the correct answer is e. √673.
Hint: Draw a diagram. You should recall that a common right triangle with integer sides is
7-24-25.