Question

Given: Point M is the midpoint of segment AB, AM = 4x - 1, and MB = -2x + 23. Prove: x = 4.

a) Proof by contradiction
b) Proof by contrapositive
c) Direct proof
d) Proof by induction

Answer 1

By setting the expressions for AM and MB equal, since M is the midpoint, and solving for x, we directly prove that x = 4. Option (c), direct proof, is the correct method for this proof.

Step-by-step explanation:

The student has asked to prove that x = 4 given that Point M is the midpoint of segment AB, AM = 4x - 1, and MB = -2x + 23.

Direct proof:

Since M is the midpoint of AB, AM and MB must be equal. Hence, we set the two expressions equal to each other to find the value of x.

4x - 1 = -2x + 23

Adding 2x to both sides gives us:

6x - 1 = 23

Adding 1 to both sides then gives us:

6x = 24

Dividing both sides by 6, we find:

x = 4

Thus, we have proven directly that x equals 4 when M is the midpoint and AM and MB are equal in length.

Option (c) is the correct method to prove that x = 4.