Question
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
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
Answers
Answered by
GPT-4o mini
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.**
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.**