Question

unit 1 lesson 1: line segments
find the measure of each segment.

1. (overline{ps}) 2. (overline{ef}) 3. (overline{jl})

find the value of the variable and (yz) if (y) is between (x) and (z).

1. (xy = 11,yz = 4c,xz = 83) 5. (xy = 6b,yz = 8b,xz = 175)
2. (xy = 4w,yz = 5w,xz = 12w - 8) 7. (xy = 3a - 4,yz = 6a + 2,xz = 5a+22)

Explanation:

Step1: Recall segment - addition postulate

If a point \(Y\) is between \(X\) and \(Z\), then \(XY + YZ=XZ\).

Step2: Solve for \(c\) in problem 4

Given \(XY = 11\), \(YZ = 4c\), \(XZ = 83\). Substitute into the segment - addition postulate: \(11+4c = 83\).
Subtract 11 from both sides: \(4c=83 - 11=72\).
Divide both sides by 4: \(c=\frac{72}{4}=18\). Then \(YZ = 4c=4\times18 = 72\).

Step3: Solve for \(b\) in problem 5

Given \(XY = 6b\), \(YZ = 8b\), \(XZ = 175\). Substitute into the segment - addition postulate: \(6b + 8b=175\).
Combine like - terms: \(14b=175\).
Divide both sides by 14: \(b=\frac{175}{14}=12.5\). Then \(YZ = 8b=8\times12.5 = 100\).

Step4: Solve for \(w\) in problem 6

Given \(XY = 4w\), \(YZ = 5w\), \(XZ = 12w−8\). Substitute into the segment - addition postulate: \(4w + 5w=12w−8\).
Combine like - terms: \(9w=12w−8\).
Subtract \(12w\) from both sides: \(9w-12w=-8\), so \(- 3w=-8\).
Divide both sides by \(-3\): \(w=\frac{8}{3}\). Then \(YZ = 5w=5\times\frac{8}{3}=\frac{40}{3}\).

Step5: Solve for \(a\) in problem 7

Given \(XY = 3a - 4\), \(YZ = 6a + 2\), \(XZ = 5a+22\). Substitute into the segment - addition postulate: \((3a - 4)+(6a + 2)=5a+22\).
Combine like - terms: \(3a-4 + 6a+2=5a+22\), \(9a-2 = 5a+22\).
Subtract \(5a\) from both sides: \(9a-5a-2=22\), \(4a-2 = 22\).
Add 2 to both sides: \(4a=22 + 2=24\).
Divide both sides by 4: \(a = 6\). Then \(YZ = 6a+2=6\times6+2=38\).

Step6: Solve for segment lengths in 1 - 3

For 1. \(PS=PQ + QR+RS\), if \(PQ = 5.8\) mm and \(QR = 3.7\) mm, then \(PR=PQ + QR=5.8+3.7 = 9.5\) mm.
For 2. \(EF=EG+GF\), if \(EG = 2.5\) in and \(GF = 2.8\) in, then \(EF=2.5 + 2.8=5.3\) in.
For 3. \(JL=JK+KL\), if \(JK = 0.75\) cm and \(KL = 0.25\) cm, then \(JL=0.75 + 0.25 = 1\) cm.

Answer:

1. \(PR = 9.5\) mm
2. \(EF = 5.3\) in
3. \(JL = 1\) cm
4. \(c = 18\), \(YZ = 72\)
5. \(b = 12.5\), \(YZ = 100\)
6. \(w=\frac{8}{3}\), \(YZ=\frac{40}{3}\)
7. \(a = 6\), \(YZ = 38\)