Write a two-column proof. Start with finding the value of x and express the properties that justify each step.
Diagram:
Proof:
Statements
Reasons
1.
AP=2x+3, PB=3x+1/2, AB=10.5
1.
Given
2.
AP+PB=AB
2.
Definition of a Segment
3.
2x+3+ 3x+1/2 =10.5
3.
Substitution Property of Equality
4.
2(2x+3+3x+1/2) =2* 10.5
4.
Multiplication Property of Equality
5.
2(2x+3+3x+1/2) =21
5.
Substitution Property of Equality
6.
2*2x+2*3+2*3x+1/2 =21
6.
Distributive Property
7.
4x+6+3x+1=21
7.
Substitution Property of Equality
8.
7x+7=21
8.
Substitution Property of Equality
9.
7x+7-7=21-7
9.
Subtraction Property of Equality
10.
7x=14
10.
Substitution Property of Equality
11.
x=2
11.
Division Property of Equality
12.
AP=2(2)+3
12.
Substitution Property of Equality
13.
AP=4+3
13.
Substitution Property of Equality
14.
AP=7
14.
Substitution Property of Equality
15.
AP/AB=7/10.5
15.
Division Property of Equality
16.
AP/AB=2/3
16.
Substitution Property of Equality
Practice makes perfect
Let's draw a diagram based on the given information.
Based on the given information, and our figure, we can write a two-column proof.
Given:& AP=2x+3, PB=3x+1/2,
&AB=10.5
Prove:& AP/AB=2/3
Writing the Proof
To start a two-column proof, we state the given information.
Given
AP=2x+3, PB=3x+1/2, AB=10.5
Next, we will write an equation using the definition of a segment. A segment is defined as the part of a line connecting two endpoints and all of the points located between the endpoints.
Definition of a Segment
AP+PB=AB
Note that there is a Postulate that tells us specifically this equation, it is called the Segment Addition Postulate and you will learn about it in the next section of the book!
Solving for x
Now, we will use the Substitution Property of Equality to substitute the given expressions into our equation. Then we can solve for x.
Finally, we can show that APAB= 23. To do this, we can substitute 7 for AP and 10.5 for AB using the Substitution Property of Equality.
Substitution Property of Equality
AP/AB=7/10.5
We will simplify the quotient by once again using the Substitution Property of Equality.
Substitution Property of Equality
AP/AB=2/3
Combining the Steps
Combining all of the above steps, we can construct our two-column proof.
