Question

this is a 2 - page document! 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. rs = st, 6x + 1, x + 7. 4. if df = 9x - 39, find ef. de = ef, 47, 3x + 10. 5. if uw = 6x - 35, find uw. 19, 4x - 20. 6. if hj = 7x - 27, find the value of x. 3x - 5, x - 1. 7. if bd = 7x - 10, bc = 4x - 29, and cd = 5x - 9, find each value. 8. if \\(\overline{bd}\cong\overline{bc}\\), bd = 5x - 26, bc = 2x + 1, and ac = 43, find ab.

Explanation:

Step1: For question 1

Since $LN = LM+MN$, substituting $LM = 22$ and $MN = 15$, we get $LN=22 + 15$.
$LN=37$

Step2: For question 2

Since $LN = LM+MN$, then $MN=LN - LM$. Substituting $LN = 54$ and $LM = 31$, we have $MN=54 - 31$.
$MN = 23$

Step3: For question 3

Since $RS=ST$ and $RT=RS + ST$, and $RT = 36$, also $RS=6x + 1$, $ST=x + 7$. Then $6x+1+x + 7=36$. Combining like - terms gives $7x+8 = 36$. Subtracting 8 from both sides: $7x=36 - 8=28$. Dividing both sides by 7, $x = 4$.

Step4: For question 4

Since $DE = EF$, and $DE = 47$, $EF=3x + 10$, then $3x+10 = 47$. Subtracting 10 from both sides: $3x=47 - 10 = 37$. Dividing both sides by 3, $x=\frac{37}{3}$, and $EF = 47$.

Step5: For question 5

Since $UW=UV+VW$, $UV = 19$, $VW=4x - 20$, and $UW=6x - 35$. Then $19+4x - 20=6x - 35$. Combining like - terms: $4x - 1=6x - 35$. Subtracting $4x$ from both sides: $-1=2x - 35$. Adding 35 to both sides: $2x=34$, $x = 17$. Then $UW=6\times17-35=102 - 35 = 67$.

Step6: For question 6

Since $HJ=HI+IJ$, $HJ = 7x - 27$, $HI=3x - 5$, $IJ=x - 1$. Then $3x-5+x - 1=7x - 27$. Combining like - terms: $4x-6=7x - 27$. Subtracting $4x$ from both sides: $-6 = 3x - 27$. Adding 27 to both sides: $3x=21$, $x = 7$.

Step7: For question 7

Since $BD=BC + CD$, $BD = 7x - 10$, $BC=4x - 29$, $CD=5x - 9$. Then $4x-29+5x - 9=7x - 10$. Combining like - terms: $9x-38=7x - 10$. Subtracting $7x$ from both sides: $2x-38=-10$. Adding 38 to both sides: $2x=28$, $x = 14$. Then $BC=4\times14 - 29=56 - 29 = 27$, $CD=5\times14 - 9=70 - 9 = 61$, $BD=7\times14 - 10=98 - 10 = 88$.

Step8: For question 8

Since $\overline{BD}\cong\overline{BC}$, $BD = 5x - 26$, $BC=2x + 1$, then $5x-26=2x + 1$. Subtracting $2x$ from both sides: $3x-26=1$. Adding 26 to both sides: $3x=27$, $x = 9$. Then $BC=2\times9 + 1=19$. Since $AC = 43$, and $AC=AB + BC$, then $AB=AC - BC$. So $AB=43 - 19=24$.

Answer:

1. $LN = 37$
2. $MN = 23$
3. $x = 4$
4. $EF = 47$
5. $UW = 67$
6. $x = 7$
7. $x = 14$, $BC = 27$, $CD = 61$, $BD = 88$
8. $AB = 24$