Math  /  Geometry

QuestionLine ss is the perpendicular bisector of JKundefined\widetilde{J K}. If line s intersects JK\overline{J K} at point L , which of the following statements must be true?
Check all that apply. A. Point L is the midpoint of JK B. Line s intersects JK\overline{\mathrm{JK}} at a 180180^{\circ} angle C. JLKLJ L K L D. Line s is parallel to JK E. Line s is perpendicular to JK\overline{J K}

Studdy Solution

STEP 1

1. Line ss is the perpendicular bisector of JKundefined\widetilde{JK}.
2. Point LL is the point of intersection of line ss and JK\overline{JK}.
3. We need to determine which statements must be true based on the properties of perpendicular bisectors and the given conditions.

STEP 2

1. Understand the properties of a perpendicular bisector.
2. Determine the geometric relationships at point LL.
3. Evaluate each given statement (A through E) based on the properties and relationships identified.

STEP 3

A perpendicular bisector of a line segment is a line that is perpendicular to the segment and divides it into two equal parts.

STEP 4

Since line ss is the perpendicular bisector of JKundefined\widetilde{JK}, it must intersect JK\overline{JK} at its midpoint.

STEP 5

The intersection point LL is, therefore, the midpoint of JK\overline{JK}.

STEP 6

Determine the angle at which line ss intersects JK\overline{JK}.

STEP 7

Since line ss is perpendicular to JK\overline{JK}, the angle of intersection at point LL is 9090^\circ, not 180180^\circ.

STEP 8

Evaluate the geometric relationship involving points JJ, LL, and KK.

STEP 9

Point LL is equidistant from points JJ and KK since it is the midpoint. However, JLJL and KLKL are not necessarily equal to each other unless specified.

STEP 10

Check whether line ss is parallel to JK\overline{JK}.

STEP 11

Line ss is perpendicular to JK\overline{JK} by definition, so it cannot be parallel to JK\overline{JK}.

STEP 12

Summarize the evaluation of each statement based on the properties of the perpendicular bisector and the given conditions.

STEP 13

Evaluate statement A: Point L is the midpoint of JK. This statement must be true because line ss is the perpendicular bisector of JK\overline{JK}.

STEP 14

Evaluate statement B: Line s intersects JK\overline{JK} at a 180180^\circ angle. This statement is false because line ss intersects JK\overline{JK} at a 9090^\circ angle.

STEP 15

Evaluate statement C: JL=KLJL = KL. This statement is true because point LL is the midpoint of JK\overline{JK}, making JLJL and KLKL equal.

STEP 16

Evaluate statement D: Line s is parallel to JK. This statement is false because line ss is perpendicular to JK\overline{JK}, not parallel.

STEP 17

Evaluate statement E: Line s is perpendicular to JK\overline{JK}. This statement is true because line ss is defined as the perpendicular bisector of JK\overline{JK}.

STEP 18

The correct statements are: A. Point L is the midpoint of JK C. JL=KLJL = KL E. Line s is perpendicular to JK\overline{JK}

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