Question

use the diagram below to answer questions 1 and 2.

1. if lm = 22 and mn = 15, find ln.
2. if ln = 54 and lm = 31, find mn.
3. if rt = 36, find the value of x.
4. if df = 9x - 39, find ef.
5. if uw = 6x - 35, find uw.
6. if hj = 7x - 27, find the value of x.
7. m is the mid - point of pq. pm = 5x + 8 and pq = 76, find x.
8. m is the mid - point of pq. pm = 9x + 3 and mq = 11x - 17, find x and pq.
9. m is the mid - point of pq. mq = 5x + 8 and pq = 6x + 56, find pm.
10. de bisects ab at c. if ac = 3x - 5 and cb = 5x + 13, find ab.

use the figure on the board for questions a - g.
a. name the intersection of plane r and line pq.
b. name three non - collinear points.
c. name a segment containing point s.
d. give another name for line sp.
e. name a line containing point q.
f. name a point coplanar to point t.
g. give another name for plane r.

Explanation:

Step1: For question 1

Use the segment - addition postulate \(LN = LM+MN\). Given \(LM = 22\) and \(MN = 15\), then \(LN=22 + 15\).

Step2: Calculate \(LN\)

\(LN=37\)

Step3: For question 2

Use the segment - addition postulate \(LN=LM + MN\), so \(MN=LN - LM\). Given \(LN = 54\) and \(LM = 31\), then \(MN=54 - 31\).

Step4: Calculate \(MN\)

\(MN = 23\)

Step5: For question 3

Since \(RT=RS+ST\) and \(RT = 36\), \(RS = 4x + 1\), \(ST=x + 7\), we have the equation \(4x+1+x + 7=36\).

Step6: Combine like - terms

\(5x+8 = 36\).

Step7: Solve for \(x\)

Subtract 8 from both sides: \(5x=36 - 8=28\), then \(x=\frac{28}{5}=5.6\)

Step8: For question 4

Since \(DF=DE + EF\), \(DF = 9x-39\), \(DE = 47\), \(EF = 3x + 10\), we have the equation \(9x-39=47+3x + 10\).

Step9: Combine like - terms

\(9x-3x=47 + 10+39\), \(6x=96\).

Step10: Solve for \(x\)

\(x = 16\), then \(EF=3x + 10=3\times16+10=58\)

Step11: For question 5

Since \(UW=UV+VW\), \(UW = 6x-35\), \(UV = 19\), \(VW = 4x-20\), we have the equation \(6x-35=19+4x-20\).

Step12: Combine like - terms

\(6x-4x=19-20 + 35\), \(2x=34\).

Step13: Solve for \(x\)

\(x = 17\), then \(UW=6x-35=6\times17-35=67\)

Step14: For question 6

Since \(HJ=HI+IJ\), \(HJ = 7x-27\), \(HI = 3x-5\), \(IJ=x - 1\), we have the equation \(7x-27=3x-5+x - 1\).

Step15: Combine like - terms

\(7x-(3x + x)=-5-1 + 27\), \(7x-4x=21\), \(3x=21\).

Step16: Solve for \(x\)

\(x = 7\)

Step17: For question 7

If \(M\) is the mid - point of \(\overline{PQ}\), then \(PM=\frac{1}{2}PQ\). Given \(PM = 5x+8\), \(PQ = 76\), we have \(5x+8=\frac{76}{2}=38\).

Step18: Solve for \(x\)

Subtract 8 from both sides: \(5x=38 - 8=30\), \(x = 6\)

Step19: For question 8

If \(M\) is the mid - point of \(\overline{PQ}\), then \(PM = MQ\). Given \(PM = 9x+3\), \(MQ = 11x-17\), we have \(9x+3=11x-17\).

Step20: Solve for \(x\)

\(17 + 3=11x-9x\), \(2x=20\), \(x = 10\). Then \(PQ=PM+MQ=9x+3+11x-17=20x-14=20\times10-14 = 186\)

Step21: For question 9

If \(M\) is the mid - point of \(\overline{PQ}\), then \(PM = MQ=\frac{1}{2}PQ\). Given \(MQ = 5x+8\), \(PQ = 6x+56\), we have \(2(5x+8)=6x+56\).

Step22: Expand the left - hand side

\(10x+16=6x+56\).

Step23: Solve for \(x\)

\(10x-6x=56 - 16\), \(4x=40\), \(x = 10\). Then \(PM=MQ=5x+8=5\times10+8=58\)

Step24: For question 10

If \(\overline{DE}\) bisects \(\overline{AB}\) at \(C\), then \(AC = CB\). Given \(AC = 3x-5\), \(CB = 5x+13\), we have \(3x-5=5x+13\).

Step25: Solve for \(x\)

\(3x-5x=13 + 5\), \(-2x=18\), \(x=-9\). Then \(AC=3x-5=3\times(-9)-5=-32\), \(CB = 5x+13=5\times(-9)+13=-32\), \(AB=AC + CB=-32+( - 32)=-64\) (distance is non - negative in geometric sense, we take the absolute value \(|AB| = 64\))

Answer:

1. \(LN = 37\)
2. \(MN = 23\)
3. \(x = 5.6\)
4. \(EF = 58\)
5. \(UW = 67\)
6. \(x = 7\)
7. \(x = 6\)
8. \(x = 10\), \(PQ = 186\)
9. \(PM = 58\)
10. \(AB = 64\)