Question

points a, b, c, and d are collinear and positioned in that order. find the length indicated.

1. (bc = x + 59), (ad = 172), (ac = 6x + 263), and (bd = x + 154). find (bd).
2. find (ab) if (bc = 58), (ad = 4x + 391), (ab = 9x + 679), and (cd = 734+10x)

Explanation:

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Step1: Use the collinear - point property

Since \(A\), \(B\), \(C\), \(D\) are collinear and in order, \(AC=AB + BC\) and \(AD=AB + BC+CD\). Also, \(AC + CD=AD\) and \(BC+CD = BD\). We know that \(AC=6x + 263\), \(BC=x + 59\), and \(BD=x + 154\). Since \(AC=AB + BC\) and \(AD=AB + BD\), we can use the fact that \(AD=AC+(BD - BC)\). Substitute the given expressions:
\[172=(6x + 263)+((x + 154)-(x + 59))\]

Step2: Simplify the right - hand side of the equation

First, simplify \((x + 154)-(x + 59)=x + 154 - x-59 = 95\). Then the equation becomes \(172=6x + 263+95\). Combine like terms: \(172=6x+358\).
\[6x=172 - 358=-186\]
\[x=- 31\]

Step3: Find the length of \(BD\)

Substitute \(x = - 31\) into the expression for \(BD\). Given \(BD=x + 154\), then \(BD=-31 + 154 = 123\)

Step1: Use the collinear - point property

Since \(A\), \(B\), \(C\), \(D\) are collinear and in order, \(AD=AB + BC+CD\). Substitute the given expressions: \(4x + 391=(9x + 679)+58+(734 + 10x)\)

Step2: Combine like terms

\[4x+391=9x + 679+58+734 + 10x\]
\[4x+391=29x + 1471\]

Step3: Solve for \(x\)

\[29x-4x=391 - 1471\]
\[25x=-1080\]
\[x = - 43.2\]

Step4: Find the length of \(AB\)

Substitute \(x=-43.2\) into the expression for \(AB\). Given \(AB = 9x+679\), then \(AB=9\times(-43.2)+679=-388.8 + 679 = 290.2\)

Answer:

\(BD = 123\)

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