Question

In the figure to the right point O is the midpoint of segment

AB
. Through point O is drawn a line
OX
.perpendicular to
AB
. Prove that any point on line
OX
is equidistant from A and B.

Answer 1

See proof below

Explanation:

Point O is the midpoint of segment AB. Through point O is drawn a line OX perpendicular to AB. Consider two triangles AOX and BOX. These triangles are right triangles, because OX is perpendicular to AB. In these triangles:

• OX=OX (reflexive property of congruence);
• AO=OB (by the definition of the midpoint O).

Thus,
`\triangle AOX\cong \triangle BOX` by LL theorem.

Congruent triangles have congruent corresponding sides, then AX=BX. This means that point X is equidistant from points A and B.