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:
-
Identify the tangent segments:
- and are tangent segments drawn from the external point to the circle.
-
Apply the tangent-segment theorem:
- Since , both tangent segments from are equal in length.
-
Given values:
- (radius of the circle),
- (distance from the center to the external point ).
-
Use the Pythagorean theorem:
- In , .
- Substituting the known values:
-
Conclude:
- Since , we find:
Final Answer:
Let me know if you'd like further clarification!
Related Questions:
- How does the tangent-segment theorem work for other shapes?
- What are the properties of tangents to a circle?
- Can the Pythagorean theorem be applied in non-right triangles?
- What if the radius was longer than ? Is this setup possible?
- 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