Question

4-1 & 4-2 intro to segments and angles

b is the midpoint of segment ac, ( ab = x + 7 ), and ( ac = 5x - 7 ).

select the geometric set up and the algebraic set up that would be used to determine the length of ( bc ).

( square ) ( ab = bc ) and ( ab + bc = ac )

( square ) ( ab = ac ) and ( ab + ac = bc )

( square ) ( 2(x + 7) = 5x - 7 )

( square ) ( x + 7 = 5x - 7 )

( square ) ( x + 7 + 5x - 7 = bc )

( square ) ( 2(5x - 7) = x + 7 )

Explanation:

Step1: Recall Midpoint Properties

Since \( B \) is the midpoint of \( AC \), by the definition of a midpoint, \( AB = BC \) and \( AB + BC = AC \) (because the midpoint divides the segment into two equal parts, so the sum of the two parts equals the whole segment).

Step2: Translate to Algebra

From \( AB = BC \) and \( AB + BC = AC \), substitute \( BC \) with \( AB \) (since \( AB = BC \)). So \( AB + AB = AC \), which simplifies to \( 2AB = AC \). Given \( AB = x + 7 \) and \( AC = 5x - 7 \), substitute these into the equation: \( 2(x + 7)=5x - 7 \).

Answer:

Geometric Set Up: \( AB = BC \) and \( AB + BC = AC \)
Algebraic Set Up: \( 2(x + 7)=5x - 7 \)