JL¯¯¯¯¯¯¯ is a perpendicular bisector of KM¯¯¯¯¯¯¯¯¯¯ . Hallie locates point H along line segment JL¯¯¯¯¯¯¯ . If the length of KH¯¯¯¯¯¯¯¯¯ is given by 6y+3 and the length of HM¯¯¯¯¯¯¯¯¯¯ is given by 2y+7 , what is the length in units of segment HM¯¯¯¯¯¯¯¯¯¯ ? (1 point)

Question

JL¯¯¯¯¯¯¯  is a perpendicular bisector of  KM¯¯¯¯¯¯¯¯¯¯ . Hallie locates point  H  along line segment  JL¯¯¯¯¯¯¯ . If the length of  KH¯¯¯¯¯¯¯¯¯  is given by  6y+3  and the length of  HM¯¯¯¯¯¯¯¯¯¯  is given by  2y+7 , what is the length in units of segment  HM¯¯¯¯¯¯¯¯¯¯ ?  (1 point)
Responses

9
9

1
1

8
8

18

GPT-4o mini · Answer

Since $ JL $ is the perpendicular bisector of $ KM $, it means that $ KH $ is equal to $ HM $. Therefore, we can set the two expressions for the lengths equal to each other: $$ KH = HM $$ Given that $ KH = 6y + 3 $ and $ HM = 2y + 7 $, we set up the equation: $$ 6y + 3 = 2y + 7 $$ Now, we'll solve for $ y $: 1. Subtract $ 2y $ from both sides: $$ 4y + 3 = 7 $$ 2. Subtract 3 from both sides: $$ 4y = 4 $$ 3. Divide both sides by 4: $$ y = 1 $$ Now that we have the value of $ y $, we can find the length of segment $ HM $: Substituting $ y = 1 $ into the expression for $ HM $: $$ HM = 2y + 7 = 2(1) + 7 = 2 + 7 = 9 $$ Thus, the length of segment $ HM $ is $ \boxed{9} $.