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 figure is missing so I attached a helping figure
Answer:
Line segment ST is congruent to line segment UT
Step-by-step explanation:
From the attached figure
∵ ST and UT are tangents to circle K at points S and U
∵ SK and UK are radii in the circle K
- The tangent is perpendicular to the radius at the point of tangent
∴ ST ⊥ KS ⇒ at point S
∴ m∠KST = 90°
∴ UT ⊥ KU ⇒ at point U
∴ m∠KUT = 90°
∴ m∠KST = m∠KUT
In the two triangles KST and KUT
∵ KS = KU ⇒ radii
∵ m∠KST = m∠KUT ⇒ proved
∵ KT is a common side in the two triangles
- That means the two triangles are congruent by HL postulate
of congruence (hypotenuse and leg of right triangle)
∴ Δ KST ≅ KUT ⇒ HL postulate of congruence
- By using the result of congruence
∴ ST = UT
Line segment ST is congruent to line segment UT
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