Questions: If BD = 7x - 10, BC = 4x - 29, and CD = 5x - 9, find each value. x = BC = 27/8 CD = 8.8 BD = 61 If BD is congruent to BC, BD = 5x - 26, BC = 2x + 1, and AC = 43, find AB.

If BD = 7x - 10, BC = 4x - 29, and CD = 5x - 9, find each value.
x = 
BC = 27/8 
CD = 8.8 
BD = 61
If BD is congruent to BC, BD = 5x - 26, BC = 2x + 1, and AC = 43, find AB.
Transcript text: If $B D=7 x-10, B C=4 x-29$, and $C D=5 x-9$, find each value. \[ \begin{aligned} x & = \\ B C & =\frac{27}{8} \\ C D & =8.8 \\ B D & =61 \end{aligned} \] If $\overline{B D} \cong \overline{B C}, B D=5 x-26, B C=2 x+1$, and $A C=43$, find $A B$.
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Solution

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Solution Steps

Step 1: Set up the equations

Given: \[ BD = 7x - 10 \] \[ BC = 4x - 29 \] \[ CD = 5x - 9 \]

Since \( BD = BC + CD \), we can write: \[ 7x - 10 = (4x - 29) + (5x - 9) \]

Step 2: Simplify the equation

Combine like terms on the right side: \[ 7x - 10 = 4x - 29 + 5x - 9 \] \[ 7x - 10 = 9x - 38 \]

Step 3: Solve for \( x \)

Isolate \( x \) by moving terms involving \( x \) to one side and constants to the other: \[ 7x - 9x = -38 + 10 \] \[ -2x = -28 \] \[ x = 14 \]

Step 4: Find the values of \( BC \), \( CD \), and \( BD \)

Substitute \( x = 14 \) back into the expressions for \( BC \), \( CD \), and \( BD \): \[ BC = 4x - 29 = 4(14) - 29 = 56 - 29 = 27 \] \[ CD = 5x - 9 = 5(14) - 9 = 70 - 9 = 61 \] \[ BD = 7x - 10 = 7(14) - 10 = 98 - 10 = 88 \]

Final Answer

\[ x = 14 \] \[ BC = 27 \] \[ CD = 61 \] \[ BD = 88 \]

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