Question

use the figure to the right to answer exercises 48 - 50. hint: re - draw the diagram below and fill in the given measures. 3. if no = 17 and np = 5x - 6, find the value of x. then find np and op. 4. if ro = 6 + x and oq = 2x + 1, find the value of x. then find ro, oq, and rq. 5. if no = 3x + 4 and np = 10x - 10, find the value of x. then find no, np, and op.

Explanation:

3.

Step1: Set up the equation

Since the line segments are related (it seems \(NP = 2NO\) from the work - assumption of mid - point or equal segments), we set \(5x−6 = 2\times17\). So \(5x−6=34\).

Step2: Solve for x

Add 6 to both sides of the equation \(5x−6 = 34\). We get \(5x=34 + 6\), so \(5x=40\). Then divide both sides by 5: \(x=\frac{40}{5}=8\).

Step3: Find NP

Substitute \(x = 8\) into the expression for \(NP\). \(NP=5x−6=5\times8−6=40 - 6=34\).

Step4: Find OP

Assuming \(NO = OP\) (from the diagram's symmetry or mid - point property), \(OP=NO = 17\).

Step1: Set up the equation

Assuming \(RO=OQ\) (from the diagram's property, likely mid - point), we set \(6 + x=2x + 1\).

Step2: Solve for x

Subtract \(x\) from both sides: \(6=2x + 1−x\), so \(6=x + 1\). Then subtract 1 from both sides: \(x=6 - 1=5\).

Step3: Find RO

Substitute \(x = 5\) into the expression for \(RO\). \(RO=6 + x=6+5 = 11\).

Step4: Find OQ

Substitute \(x = 5\) into the expression for \(OQ\). \(OQ=2x + 1=2\times5+1=10 + 1=11\).

Step5: Find RQ

\(RQ=RO + OQ=11+11 = 22\).

Step1: Set up the equation

Assuming \(NP = 2NO\) (from the diagram's property), we set \(10x−10=2(3x + 4)\).

Step2: Expand the right - hand side

\(10x−10=6x+8\).

Step3: Solve for x

Subtract \(6x\) from both sides: \(10x−6x−10=6x−6x + 8\), so \(4x−10=8\). Add 10 to both sides: \(4x=8 + 10=18\). Then divide both sides by 4: \(x=\frac{18}{4}=\frac{9}{2}=4.5\).

Step4: Find NO

Substitute \(x = 4.5\) into the expression for \(NO\). \(NO=3x + 4=3\times4.5+4=13.5 + 4=17.5\).

Step5: Find NP

Substitute \(x = 4.5\) into the expression for \(NP\). \(NP=10x−10=10\times4.5−10=45 - 10=35\).

Step6: Find OP

Assuming \(NO = OP\), \(OP=NO = 17.5\).

Answer:

\(x = 8\), \(NP=34\), \(OP = 17\)

4.