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. if (bd = 7x-10), (bc = 4x - 29), and (cd = 5x - 9), find each value.
8. if (overline{bd}congoverline{bc}), (bd = 5x - 26), (bc = 2x + 1), and (ac = 43), find (ab).

Explanation:

Step1: Analyze question 1

Since $LN = LM+MN$, given $LM = 22$ and $MN = 15$, then $LN=22 + 15$.
$LN=37$

Step2: Analyze question 2

Since $LN=LM + MN$, then $MN=LN - LM$. Given $LN = 54$ and $LM = 31$, so $MN=54-31$.
$MN = 23$

Step3: Analyze question 3

Since $RT=RS+ST$, and $RT = 36$, $RS=6x + 1$, $ST=x + 7$. Then $6x+1+x + 7=36$, which simplifies to $7x+8 = 36$. Subtract 8 from both sides: $7x=36 - 8=28$, divide both sides by 7, $x = 4$.

Step4: Analyze question 4

Since $DF=DE+EF$, and $DF = 9x-39$, $DE = 47$, $EF=3x + 10$. Then $9x-39=47+3x + 10$. Combine like - terms: $9x-3x=47 + 10+39$, $6x=96$, $x = 16$. Substitute $x = 16$ into $EF=3x + 10$, $EF=3\times16+10=58$.

Step5: Analyze question 5

Since $UW=UV+VW$, and $UV = 19$, $VW=4x-20$, $UW=6x-35$. Then $6x-35=19+4x-20$, $6x-4x=19-20 + 35$, $2x=34$, $x = 17$. Substitute $x = 17$ into $UW=6x-35$, $UW=6\times17-35=67$.

Step6: Analyze question 6

Since $HJ=HI+IJ$, and $HJ = 7x-27$, $HI=3x-5$, $IJ=x - 1$. Then $7x-27=3x-5+x - 1$, $7x-3x-x=-5-1 + 27$, $3x=21$, $x = 7$.

Step7: Analyze question 7

Since $BD=BC+CD$, and $BD = 7x-10$, $BC=4x-29$, $CD=5x-9$. Then $7x-10=4x-29+5x-9$, $7x-10=9x-38$, $9x-7x=38 - 10$, $2x=28$, $x = 14$. Substitute $x = 14$: $BC=4\times14-29=27$, $CD=5\times14-9=61$, $BD=7\times14-10=88$.

Step8: Analyze question 8

Since $\overline{BD}\cong\overline{BC}$, then $5x-26=2x + 1$, $5x-2x=1 + 26$, $3x=27$, $x = 9$. $BC=2x + 1=2\times9+1=19$. Since $AC = 43$, then $AB=AC - BC=43-19=24$.

Answer:

1. $LN = 37$
2. $MN = 23$
3. $x = 4$
4. $EF = 58$
5. $UW = 67$
6. $x = 7$
7. $x = 14$, $BC = 27$, $CD = 61$, $BD = 88$
8. $AB = 24$