Segment Addition Postulate

1.

E is the midpoint between D and F. If DE = 3x + 2, EF = 18, and DF = 5x, find the value of x.

10

15

3

5

Since E is the midpoint between D and F, we can set up an equation using the midpoint formula. According to the formula, the sum of the distances from E to D and E to F should be equal. So, DE + EF = DF. Substituting the given values, we get (3x + 2) + 18 = 5x. Simplifying the equation, we have 3x + 20 = 5x. By subtracting 3x from both sides, we get 20 = 2x. Dividing both sides by 2, we find x = 10. Therefore, the value of x is 10.

Explanation

Since E is the midpoint between D and F, we can set up an equation using the midpoint formula. According to the formula, the sum of the distances from E to D and E to F should be equal. So, DE + EF = DF. Substituting the given values, we get (3x + 2) + 18 = 5x. Simplifying the equation, we have 3x + 20 = 5x. By subtracting 3x from both sides, we get 20 = 2x. Dividing both sides by 2, we find x = 10. Therefore, the value of x is 10.

2.

E is the midpoint between D and F. If DE = 3x + 2, EF = 18, and DF = 5x, find the value of DE.

10

32

30

25

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Explanation

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3.

E is the midpoint between D and F. If DE = 3x + 2, EF = 18, and DF = 5x, find the value of DF.

10

32

45

50

Since E is the midpoint between D and F, we can use the midpoint formula to find the value of DF. The midpoint formula states that the coordinates of the midpoint (E) are the average of the coordinates of the endpoints (D and F). Therefore, we can set up the equation (DE + EF) / 2 = DF and substitute the given values. (3x + 2 + 18) / 2 = 5x. Simplifying this equation, we get 21 / 2 = 5x. Solving for x, we find that x = 21/10. Substituting this value back into the equation for DF, we get DF = 5(21/10) = 105/10 = 10.5. However, since the answer choices are all integers, the nearest whole number to 10.5 is 11, so the value of DF is 11.

Explanation

Since E is the midpoint between D and F, we can use the midpoint formula to find the value of DF. The midpoint formula states that the coordinates of the midpoint (E) are the average of the coordinates of the endpoints (D and F). Therefore, we can set up the equation (DE + EF) / 2 = DF and substitute the given values. (3x + 2 + 18) / 2 = 5x. Simplifying this equation, we get 21 / 2 = 5x. Solving for x, we find that x = 21/10. Substituting this value back into the equation for DF, we get DF = 5(21/10) = 105/10 = 10.5. However, since the answer choices are all integers, the nearest whole number to 10.5 is 11, so the value of DF is 11.

4. What is the value of x?

8

6

2

4

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Explanation

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5. J is between H and K. HJ = 2x + 5 JK = 3x - 7 KH = 18 Find JK.

5

8

7

4

Since J is between H and K, we can determine the value of x by setting the distances between J and H and J and K equal to each other.
Therefore, 2x + 5 = 18 - (3x - 7).
Simplifying the equation gives us 5x = 20, which means x = 4.
Substituting x = 4 into JK = 3x - 7 gives us JK = 3(4) - 7 = 12 - 7 = 5.
Therefore, the correct answer is 5.

Explanation

Since J is between H and K, we can determine the value of x by setting the distances between J and H and J and K equal to each other.
Therefore, 2x + 5 = 18 - (3x - 7).
Simplifying the equation gives us 5x = 20, which means x = 4.
Substituting x = 4 into JK = 3x - 7 gives us JK = 3(4) - 7 = 12 - 7 = 5.
Therefore, the correct answer is 5.

6. M is between A and B. AM = 3 + 2x AB = 9x - 13 MB = 6x - 12 Find AB

4

23

36

13

Given that M is between A and B, we can infer that the sum of AM and MB should equal AB.
Therefore, we can set up the equation:
AM + MB = AB
(3 + 2x) + (6x - 12) = 9x - 13
Simplifying the equation, we get:
9x - 9 = 9x - 13
-9 = -13
This equation is not possible, as -9 cannot equal -13.
Therefore, there is no valid solution for AB, and the question is incomplete.

Explanation

Given that M is between A and B, we can infer that the sum of AM and MB should equal AB.
Therefore, we can set up the equation:
AM + MB = AB
(3 + 2x) + (6x - 12) = 9x - 13
Simplifying the equation, we get:
9x - 9 = 9x - 13
-9 = -13
This equation is not possible, as -9 cannot equal -13.
Therefore, there is no valid solution for AB, and the question is incomplete.

7. M is between A and B. AM = MB AM = 5x + 3 MB = 6x - 4 Find AB.

36

38

7

76

Since M is between A and B, we can set up an equation using the given information. AM = MB, and substituting the given expressions for AM and MB, we get 5x + 3 = 6x - 4. Solving for x, we find that x = 7. To find AB, we substitute x = 7 into either AM or MB. Using AM, we get AB = AM + MB = (5*7 + 3) + (6*7 - 4) = 35 + 3 + 42 - 4 = 76. Therefore, the correct answer is 76.

Explanation

Since M is between A and B, we can set up an equation using the given information. AM = MB, and substituting the given expressions for AM and MB, we get 5x + 3 = 6x - 4. Solving for x, we find that x = 7. To find AB, we substitute x = 7 into either AM or MB. Using AM, we get AB = AM + MB = (5*7 + 3) + (6*7 - 4) = 35 + 3 + 42 - 4 = 76. Therefore, the correct answer is 76.