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Point Z is equidistant from the vertices of ΔTUV.

Point Z is equidistant from the vertices of triangle T U V. Lines are drawn from point Z to the sides of the triangle to form right angles and line segments Z A, Z B, and Z C.
Which must be true?

Line segment T A is-congruent-to line segment T B
Line segment A Z is-congruent-to line segment B Z
AngleBTZ Is-congruent-to AngleBUZ
AngleTZA Is-congruent-to AngleTZB

1 answer

Given that point Z is equidistant from the vertices of triangle TUV, this means that ZA, ZB, and ZC (the perpendiculars from Z to the sides of the triangle) represent the distances from Z to the respective sides of the triangle.

Let's analyze the statements provided:

  1. Line segment TA is congruent to line segment TB: This statement is NOT necessarily true. Since Z is equidistant from the vertices of the triangle, it does not imply that the lengths TA and TB will be equal.

  2. Line segment AZ is congruent to line segment BZ: This statement is NOT necessarily true. Z being equidistant from the vertices does not imply that the segments from Z to the points where the perpendiculars meet the sides of the triangle are equal.

  3. Angle BTZ is congruent to angle BUZ: This statement is NOT necessarily true. There is no specific information given about these angles being congruent just from Z being equidistant from the vertices.

  4. Angle TZA is congruent to angle TZB: This statement IS true because both angles are formed by the same line (TZ) intersecting the lines TA and TB. In essence, they correspond to the segments connecting Z to the respective sides of the triangle and will be equal if Z is equidistant from those sides.

Therefore, the only statement that must be true is: Angle TZA is congruent to Angle TZB.