This image contains three geometry problems involving circles and triangles, where the task is to find the segment lengths, assuming tangency for the lines that appear to be tangent.

Problem 1:

• There is a circle with a triangle inscribed, and the values provided are:
  • One side of the triangle is 12.
  • Another side of the triangle is 9.
  • The hypotenuse, which lies along the radius of the circle (14).

This appears to be applying the Pythagorean Theorem.

Problem 2:

• There is a circle with a tangent line and a triangle formed outside the circle. The known values are:
  • Segment $AC = 50$,
  • Segment $AB = x - 24$.

We are likely using the property of tangents from a point outside the circle.

Problem 3:

• Another triangle is inscribed in a circle, where:
  • One side is 8,
  • Another side is 16.

This is again likely a case where the Pythagorean Theorem will be used.

Detailed Solutions:

1. Problem 1:

   • This forms a right triangle, where the sides are 12, 9, and the hypotenuse is 14. Using the Pythagorean Theorem: $12^2 + 9^2 = 14^2$ Solving will verify the tangency.

2. Problem 2:

   • Using the property of tangents, segments from a common external point are equal. Therefore, $AB = AC$, meaning: $x - 24 = 50.$ Solve for $x$.

3. Problem 3:

   • This forms another right triangle, likely, where the sides are given as 8, 16. We can check if the hypotenuse or the other length follows the Pythagorean Theorem.

Would you like me to go step-by-step for each problem or focus on any specific one?

Next Steps:

1. Do you need a full explanation of the Pythagorean Theorem applications?
2. Should I break down the tangent line property further?
3. Do you have any questions about specific geometry principles here?
4. Want to know more about tangents and circles?
5. Interested in how the external tangents relate to circles?

Tip:

Whenever you have tangents from a point outside the circle, the tangents to the circle from that point are always equal in length!