Chapter 6.2, Problem 3PSA

To determine

To prove: The given line segments $\overline{\text{RS}}$ and $\overline{\text{RT}}$ are congruent

Explanation of Solution

Given information:

A circle with center O lies in plane n

One line segment is perpendicular to plane ‘n’, i.e., $\overline{\text{RO}}\perp \text{n}$

Formula used:

Congruency means same shape and same size

Congruent Parts of congruent Triangles are Congruent

Reflexive property

If a line is perpendicular to plane than all line passing through its foot will also perpendicular to it

Perpendicular lines always form right angles

Proof:

  $\overline{\text{RO}}\perp \text{n}$........... given

  $\overline{\text{RO}}\perp \overline{\text{OS}}\text{ \& }\overline{\text{RO}}\perp \overline{\text{OT}}$........... (If a line is perpendicular to plane than all line passing through its foot will also be perpendicular to it)

  $\angle ROS=\angle ROT=90°$ ...........Perpendicular lines from right angles

  $\overline{\text{RS}}\cong \overline{\text{RT}}$ .............................. Reflexive property

  $\overline{\text{RO}}\cong \overline{\text{RO}}$.................Reflexive Property

  $\Delta \text{ROS}\cong \Delta \text{ROT}$ SAS criteria is satisfied

  $\overline{\text{RS}}\cong \overline{\text{RT}}$.................CPCTC

Hence proved