Question

read 7.1 bisectors of triangles (textbook pgs. 406-410)
perpendicular bisector a perpendicular bisector is a line, segment, or ray that is perpendicular to the given segment and passes through its midpoint. some theorems deal with perpendicular bisectors.

perpendicular bisector theorem	if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
converse of perpendicular bisector theorem	if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
circumcenter theorem	the perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle.

example 1 find the measure of ( fm ).
triangle diagram with ( fk ) as perpendicular bisector of ( gm ), ( fg = 2.8 )
( overline{fk} ) is the perpendicular bisector of ( overline{gm} ).
( fg = fm )
( 2.8 = fm )
example 2 ( overline{bd} ) is the perpendicular bisector of ( overline{ac} ). find ( x ).
triangle diagram with ( bd ) as perpendicular bisector, ( ad = 3x + 8 ), ( dc = 5x - 6 )
( ad = dc )
( 3x + 8 = 5x - 6 )
( 14 = 2x )
( 7 = x )
exercises
find each measure.

1. ( xw )

triangle diagram with perpendicular bisector, ( xy = 7.5 ), ( yw = 5 )

1. ( bf )

triangle diagram with perpendicular bisector, ( ae = 19 ), ( be ) perpendicular to ( af ), ( ab = 4.2 )

Explanation:

Problem 1: Find \( XW \)

Step 1: Identify the perpendicular bisector

The segment (let's say \( XY \) and \( XW \) related) – the perpendicular bisector implies that the two triangles are congruent or the sides are equal. Wait, looking at the diagram, the perpendicular bisector (the right angle mark) suggests that the distance from \( X \) to \( Y \) and \( X \) to \( W \) – wait, the side \( XY = 7.5 \)? Wait, no, the diagram has \( Y \) to the perpendicular bisector segment is 5, \( W \) to the perpendicular bisector segment is 5, and \( XY = 7.5 \). By the Perpendicular Bisector Theorem, if a point is on the perpendicular bisector, it is equidistant from the endpoints. So \( X \) is on the perpendicular bisector, so \( XY = XW \).

Step 2: Set \( XY = XW \)

Given \( XY = 7.5 \), so \( XW = 7.5 \).

Step 1: Identify the perpendicular bisector

\( EB \) is the perpendicular bisector of \( AF \) (since \( AB = BF \)? Wait, the diagram shows \( AE = 19 \), \( FE = 19 \), and \( EB \) is perpendicular to \( AF \) (right angle mark). By the Perpendicular Bisector Theorem, \( EB \) is the perpendicular bisector, so \( AB = BF \). Wait, \( AB = 4.2 \)? Wait, the diagram has \( A \) to \( B \) is 4.2, and \( B \) to \( F \) – since \( EB \) is the perpendicular bisector, \( AB = BF \).

Step 2: Set \( AB = BF \)

Given \( AB = 4.2 \), so \( BF = 4.2 \).

Answer:

\( 7.5 \)

Problem 2: Find \( BF \)