Question

Anyone knows what the answer to problem a and c??

Answer 1

Answers:

(a) Altitude

(b) Angle Bisector

(c) Perpendicular bisector

Explanations

• (a) The altitude goes through a vertex and is perpendicular to the opposite side of this vertex. Often "height" is used in place of "altitude".
• (b) The angle bisector splits an angle into two congruent pieces. Example: a 60 degree angle split into two 30 degree angles.
• (c) The perpendicular bisector is perpendicular to one side, and it splits that side into two equal pieces (similar to what the angle bisector does). The perpendicular bisector may or may not intersect the opposite vertex. In this case, it doesn't. If it did, then we'd also consider it a median.

Further notes:

• Intersect two or more altitudes and you get the orthocenter.
• Intersect two or more angle bisectors, and you get the incenter. This is useful to construct the incircle. It's the largest circle possible that fits inside the triangle without spilling outside the triangle.
• Intersect two or more perpendicular bisectors, and you get the circumcenter. This is useful to construct the circumcircle. It's the circle that passes through all 3 vertices of the triangle. This is the smallest circle possible that has the triangle inside the circle without spilling outside the circle.

Answer 2

Line segment ZV serves as an altitude, 05 acts as an angle bisector, and 10 functions as a perpendicular bisector. Each plays a distinct role in understanding and analyzing the geometric properties of the triangle.

(a) Line segment ZV is an altitude:

An altitude in a triangle is a line segment drawn from a vertex of the triangle perpendicular to the opposite side. In this case, line segment ZV passes through the vertex Z of the triangle XYZ and is perpendicular to the opposite side XY. Altitudes are crucial in geometry as they help us find the height of the triangle and are instrumental in various geometric and trigonometric calculations.

(b) Line segment 05 is an angle bisector:

An angle bisector is a line or line segment that divides an angle into two congruent angles. In the given triangle, line segment 05 bisects angle Q, meaning it divides angle Q into two angles of equal measure. This is a significant property, often employed in geometric constructions and trigonometric problem-solving.

(c) Line segment 10 is a perpendicular bisector:

A perpendicular bisector is a line or line segment that is perpendicular to a given line segment and divides it into two equal parts. In this case, line segment 10 is perpendicular to side FG and divides it into two equal segments. Notably, the perpendicular bisector doesn't necessarily intersect the opposite vertex (in this case, H), which distinguishes it from a median.