Circle K is shown. Tangents S T and U T intersect at point T outside of the circle. A line is drawn from point T to point R on t
he opposite side of the circle. It goes through center point K. Lines are drawn from points S and U to center point K.
Line segment ST is congruent to which line segment?
Line segment ST is congruent to which line segment?
2 answers:
The diagram is not available, so I included a supplementary figure.
Response:
The segments ST and UT are equal in length.
Detailed explanation:
As seen in the additional figure
∵ ST and UT act as tangents to circle K at points S and U respectively
∵ SK and UK are the radii of circle K
- A tangent is perpendicular to the radius at the point where it touches the circle.
Thus, ST ⊥ KS at point S
Thus, m∠KST = 90°
Thus, UT ⊥ KU at point U
Thus, m∠KUT = 90°
Therefore, m∠KST = m∠KUT
In triangles KST and KUT
∵ KS = KU because they are both radii
∵ m∠KST = m∠KUT confirms the equality
∵ KT is common to both triangles
- Therefore, the triangles are congruent according to the HL criterion
∴ Δ KST ≅ KUT as per the HL criterion
- Thus, from the congruency result
∴ ST = UT
The segments ST and UT are equal in length.
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