Part A: MUST SHOW ALL WORK FOR CRBDIT: (10 pens Total
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1.
B
Given: AB = 28; M lies on the line AB
Prove: M is a midpoint.
Statements
111 M lies on AB
E-08
2 A01-11B = AB
(3) 24 4 2 + 43 - 10 - 28
162-8=28
562=36
6) х = 6
7) AM = 2(6] + 2 and
(MB = 4(6) - 10
8) AM = 14 and MB = 14
9) M is a midpoint
M
Reasons
111
21
37 3mb-riE2n Propersy
41
51
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(7) Substitution Property
8) Simplifying
191
Reasoning Bask:
•
•
Addition Pragedy of Engality
gombina Like Lares
Defiratias, of Midpoint
.
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Division Propert of Equalitar
Given
Segment Addition Postulate
Proof:
1. Given: AB = 28; M lies on the line AB
2. M lies on AB (Given)
3. A + MB = AB (Segment Addition Postulate)
4. 2 * AM + 2 = AB (Substitution Property)
5. 2 * AM + 2 = 28 (Substitution of AB = 28)
6. 2 * AM = 28 - 2 (Subtraction Property)
7. 2 * AM = 26 (Simplifying)
8. AM = 26 / 2 (Division Property)
9. AM = 13 (Simplifying)
10. MB = AB - AM (Segment Addition Postulate)
11. MB = 28 - 13 (Substitution of AB = 28 and AM = 13)
12. MB = 15 (Simplifying)
Therefore, we have proven that AM = 13 and MB = 15. Since AM = MB, we can conclude that M is the midpoint of AB.
Reasoning:
- In step 3, we use the Segment Addition Postulate which states that if M lies on AB, then the lengths of AM and MB added together equal the length of AB.
- In step 4, we substitute AB with 28.
- In step 7, we simplify the equation to solve for AM.
- In step 8, we divide both sides of the equation by 2 to solve for AM.
- In step 10, we use the Segment Addition Postulate again to find the length of MB.
- In step 12, we simplify the equation to find that MB = 15.
- Since AM = MB, we can conclude that M is the midpoint of AB.
To prove that M is a midpoint, we need to show that AM is equal in length to MB.
Given: AB = 28; M lies on the line AB
Statements:
1. M lies on AB (Given)
2. AM + MB = AB (Segment Addition Postulate)
3. 2x + 42 = 28 (Substitute AM = 2x and MB = 4x - 10)
4. 2x + 42 - 42 = 28 - 42 (Subtract 42 from both sides)
5. 2x = -14 (Simplify)
6. x = -14 / 2 (Divide both sides by 2)
7. x = -7 (Simplify)
8. AM = 2(-7) + 2 (Substitute x = -7)
9. AM = -14 + 2 (Simplify)
10. AM = -12 (Simplify)
11. MB = 4(-7) - 10 (Substitute x = -7)
12. MB = -28 - 10 (Simplify)
13. MB = -38 (Simplify)
14. AM = MB (Substitute the values of AM and MB)
15. M is a midpoint (Definition of Midpoint)
Reasons:
1. Given
2. Segment Addition Postulate
3. Substitution
4. Subtraction
5. Simplification
6. Division
7. Substitution
8. Simplification
9. Simplification
10. Simplification
11. Substitution
12. Simplification
13. Simplification
14. Substitution
15. Definition of Midpoint
The problem is asking us to prove that point M is the midpoint of line segment AB. To do this, we need to show that the lengths of AM and MB are equal.
We are given that AB = 28 units and M lies on line AB.
Now let's go through the statements and reasons provided to prove that M is a midpoint:
Statement 1: M lies on AB
Reason: Given
Statement 2: AB = AB
Reason: Reflexive Property of Equality
Statement 3: 2x = 10
Reason: Substitution Property (Using the given lengths, AB = 2x and 28 = 10)
Statement 4: x = 5
Reason: Simplifying (Dividing both sides of the equation by 2)
Statement 5: AM = 2(5) + 2 and MB = 4(5) - 10
Reason: Using the formula for finding the lengths of AM and MB
Statement 6: AM = 14 and MB = 14
Reason: Simplifying
Statement 7: AM = MB
Reason: Subtracting 14 from both sides of the equation AM - MB = 0
Statement 8: M is a midpoint
Reason: Definition of midpoint (If the lengths of AM and MB are equal, then M is a midpoint)
So, by showing that the lengths of AM and MB are equal, we have proven that M is a midpoint of line segment AB.