Question

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 the 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?

Answer 1

Answer: C) UT

Explanation:

Just took the test

Answer 2

The figure is missing so I attached a helping figure

Answer:

Line segment ST is congruent to line segment UT

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