Question

Given collinear points A, B, and C such that point B is the midpoint of segment AC. Find the length of BC given that AB = 2x + 9 and AC = 9x + 3.

Answer 1

To find the length of BC, we set up and solve the equation AC = 2(AB) using the given expressions for AB and AC. Solving for 'x' gives us x=3. Substituting 'x' back into AB or BC, we find that the length of BC is 15 units.

Step-by-step explanation:

The length of BC given collinear points A, B, and C, where B is the midpoint of AC, can be found using the given information that AB = 2x + 9 and AC = 9x + 3. Since B is the midpoint, AB = BC, so we set AB equal to BC and solve for 'x'. Once 'x' is known, substitute the value back into either AB or BC (since they are equal) to get the length of BC.

To solve for 'x':

1. Write the equation based on the information that AC is twice the length of AB: AC = 2(AB).
2. Substitute the given expressions into the equation: 9x + 3 = 2(2x + 9).
3. Simplify and solve for 'x': 9x + 3 = 4x + 18, which results in 5x = 15, so x = 3.
4. Substitute 'x' back into AB or BC: AB = 2(3) + 9 = 6 + 9 = 15.

Therefore, the length of BC is 15 units.