Math Problem Statement

In the figure below, the segments ST and SU are tangent to the circle centered at O. Given that OT = 3 and OS = 5, find SU.

Solution

To solve the problem, we use the tangent-segment theorem, which states that two tangent segments drawn from the same external point to a circle are equal in length.

Step-by-step solution:

  1. Identify the tangent segments:

    • STST and SUSU are tangent segments drawn from the external point SS to the circle.
  2. Apply the tangent-segment theorem:

    • Since ST=SUST = SU, both tangent segments from SS are equal in length.
  3. Given values:

    • OT=3OT = 3 (radius of the circle),
    • OS=5OS = 5 (distance from the center OO to the external point SS).
  4. Use the Pythagorean theorem:

    • In OST\triangle OST, OS2=OT2+ST2OS^2 = OT^2 + ST^2.
    • Substituting the known values: 52=32+ST25^2 = 3^2 + ST^2 25=9+ST225 = 9 + ST^2 ST2=16ST^2 = 16 ST=4ST = 4
  5. Conclude:

    • Since SU=STSU = ST, we find: SU=4SU = 4

Final Answer:

SU=4SU = 4

Let me know if you'd like further clarification!


Related Questions:

  1. How does the tangent-segment theorem work for other shapes?
  2. What are the properties of tangents to a circle?
  3. Can the Pythagorean theorem be applied in non-right triangles?
  4. What if the radius OTOT was longer than OSOS? Is this setup possible?
  5. How can we verify tangent lengths geometrically using constructions?

Tip: Always verify if a triangle is right-angled before applying the Pythagorean theorem!

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Properties
Pythagorean Theorem

Formulas

Pythagorean Theorem: a^2 + b^2 = c^2

Theorems

Tangent-Segment Theorem

Suitable Grade Level

Grades 8-10