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In the picture below, AG = GB

If AG = 7x+9 and GB = 12x - 6, find the length of AB

(1 point)
AB=

Question 3
For the picture below FG ¯¯¯¯¯¯¯¯¯
is a perpendicular bisector of AB¯¯¯¯¯¯¯¯
. Which points are equidistant from points A and B? Check ALL that apply

(4 points)
Responses

W
W

X
X

Y
Y

Z
Z

P
P

S
S

F
F

G
G
Question 4
Which theorem proves this?(1 point)
Responses

Vertical Angles theorem
Vertical Angles theorem

Perpendicular Bisector Theorem
Perpendicular Bisector Theorem

Converse of Alternate Interior Angles Theorem
Converse of Alternate Interior Angles Theorem

Triangle Sum Theorem
Triangle Sum Theorem

1 answer

Question 1: Finding the length of AB

Given:

  • \( AG = GB \)
  • \( AG = 7x + 9 \)
  • \( GB = 12x - 6 \)

Since \( AG = GB \), we can set the two expressions equal to each other:

\[ 7x + 9 = 12x - 6 \]

To solve for \( x \), first, rearrange the equation:

\[ 9 + 6 = 12x - 7x \] \[ 15 = 5x \] \[ x = 3 \]

Now, substitute \( x = 3 \) back into either equation to find \( AG \):

\[ AG = 7(3) + 9 = 21 + 9 = 30 \]

Since \( AG = GB \), we have:

\[ GB = 30 \]

The total length \( AB \) is the sum of \( AG \) and \( GB \):

\[ AB = AG + GB = 30 + 30 = 60 \]

Thus, the length of \( AB \) is 60.

Question 3: Points Equidistant from A and B

Since \( FG \) is the perpendicular bisector of \( AB \), by definition, any point on the perpendicular bisector is equidistant from points A and B. Therefore, any point \( P \) that lies on line \( FG \) will be equidistant from A and B.

To select the correct responses, look for the points that lie on the perpendicular bisector \( FG \):

  • Responses that are on the line \( FG \): F, G, W, X, Y, Z, and P.

Points S may or may not lie on line ( FG **, so to confirm, you need a visual or additional information.

Question 4: Which theorem proves this?

The theorem that states that any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment is called the Perpendicular Bisector Theorem.

So the correct response is: Perpendicular Bisector Theorem.