Circles Tangents Worksheets - Free Printable

This is a geometry problem involving a circle and a triangle. The key is to recognize that the triangle is a right triangle, with the right angle at the point where the side of length 16 touches the circle.

Here's the solution:

- The triangle has sides of length 8, 16, and x.
- The side of length 8 is part of a line segment that passes through the center of the circle, making it a radius.
- The side of length 16 is tangent to the circle. A fundamental property of circles is that a radius drawn to the point of tangency is perpendicular to the tangent line.
- Therefore, the angle between the side of length 8 (which is a radius) and the side of length 16 (the tangent) is 90 degrees. This makes the triangle a right triangle.

We can now apply the Pythagorean theorem:
a² + b² = c²

Where:
- a = 8
- b = 16
- c = x (the hypotenuse)

So:
8² + 16² = x²
64 + 256 = x²
320 = x²

x = √320
x = √(64 * 5)
x = 8√5

However, none of the answer choices match 8√5. Let's re-examine the diagram. It appears that the side labeled "x" is not the hypotenuse but one of the legs of the right triangle. The hypotenuse would be the side connecting the two endpoints of the 8 and 16 segments.

Let's assume the side labeled "x" is the other leg of the right triangle, and the hypotenuse is the side connecting the end of the 8-unit segment to the end of the 16-unit segment. But this doesn't fit the diagram either.

Looking again, the most logical interpretation is that the triangle is a right triangle with legs of 8 and 16, and the hypotenuse is x. But as we saw, that gives x = 8√5, which is not an option.

Perhaps the side labeled "x" is the radius, and the hypotenuse is the side from the center of the circle to the end of the 16-unit segment. In that case, the right triangle has legs of 8 and 16, and the hypotenuse is the distance from the center of the circle to the end of the 16-unit segment. But that doesn't help us find x.

Wait, let's consider that the side labeled "x" might be the entire length from the leftmost point to the center of the circle. In that case, the side of length 8 is part of that segment, and the remaining part is also labeled "x". That doesn't make sense.

Another possibility: the side labeled "x" is the radius, and the triangle has sides 8, 16, and x, with the right angle between the 8 and 16 sides. Then x is the hypotenuse, and we have:

8² + 16² = x²
64 + 256 = x²
320 = x²
x = √320 = 8√5 ≈ 17.89

Still not matching any options.

Let's look at the answer choices: 4, 10, 6, 12. These are all integers. Perhaps there's a different interpretation.

Maybe the side labeled "x" is not the hypotenuse but one of the legs, and the hypotenuse is the side from the center of the circle to the end of the 16-unit segment. But then we have a right triangle with legs 8 and x, and hypotenuse 16? That would give:

8² + x² = 16²
64 + x² = 256
x² = 192
x = √192 = 8√3 ≈ 13.86

Not matching.

Perhaps the side labeled "x" is the radius, and the triangle has sides 8, 16, and x, with the right angle between the 8 and x sides. Then 16 is the hypotenuse:

8² + x² = 16²
64 + x² = 256
x² = 192
x = √192 = 8√3 ≈ 13.86

Still not matching.

Wait, perhaps the side labeled "x" is the radius, and the side of length 8 is not a leg of the triangle but part of the line from the center to the leftmost point. In that case, the triangle has legs of length x (radius) and 16 (tangent), and the hypotenuse is the line from the center to the end of the 16-unit segment. But that doesn't help us find x.

Let's try a different approach. Maybe the side labeled "x" is the radius, and the side of length 8 is the distance from the leftmost point to the point of tangency. Then the triangle has legs of length x and 16, and the hypotenuse is the line from the center to the leftmost point, which is 8 + x.

Then we have:

x² + 16² = (8 + x)²
x² + 256 = 64 + 16x + x²
256 = 64 + 16x
192 = 16x
x = 12

That matches option D.

So the correct interpretation is:
- The side labeled "x" is the radius of the circle.
- The side of length 16 is tangent to the circle.
- The side of length 8 is the distance from the leftmost point to the point where the radius meets the tangent line.
- The hypotenuse of the right triangle is the line from the center of the circle to the leftmost point, which has length 8 + x.

Applying the Pythagorean theorem:
x² + 16² = (8 + x)²
x² + 256 = 64 + 16x + x²
256 = 64 + 16x
192 = 16x
x = 12

Therefore, the answer is 12.

D) 12