B is the midpoint of AC and D is the midpoint of CE Solve for BD given that BD = 2y + 8 and AE = 6y +12

Answer 1

Since B is the midpoint of AC, we know that AB = BC.

Similarly, since D is the midpoint of CE, we know that CD = DE.

Therefore, AE = AC + CE.

Substituting the given values:

6y + 12 = (AB + BC) + (CD + DE).

Since AB = BC and CD = DE, we can simplify this to:

6y + 12 = 2(AB + CD).

Now, we are given that BD = 2y + 8.

Since AB + BD + CD = AC, we can substitute the given values and expressions:

AB + (2y + 8) + CD = AC.

Since AB = BC and CD = DE, we can rewrite this as:

BC + (2y + 8) + DE = AC.

Therefore, AC = 2(BC + DE).

Since B is the midpoint of AC and D is the midpoint of CE, we can rewrite this as:

AC = 2(BD).

Substituting the given values and expressions:

6y + 12 = 2(2y + 8).

Simplifying this equation:

6y + 12 = 4y + 16.

Subtracting 4y from both sides:

2y + 12 = 16.

Subtracting 12 from both sides:

2y = 4.

Dividing both sides by 2:

y = 2.

Now we can substitute this value back into the equation for BD:

BD = 2(2y + 8) = 2(2(2) + 8) = 2(4 + 8) = 2(12) = 24.

Therefore, BD = 24.