Question

Find AC if BC=3x-81, AD=6x-30 AC=11x-290, and BD=115.

Answer 1

AC is equal to 68.

Step-by-step explanation:

To find AC, we need to use the information provided about the segments BC, AD, and BD. The given values are:

`\[BC = 3x - 81\]`

`\[AD = 6x - 30\]`

`\[AC = 11x - 290\]`

`\[BD = 115\]`

Firstly, we'll establish the relationship between BC and BD:

`\[BC + CD = BD\]`

`\[3x - 81 + CD = 115\]`

`\[CD = 196 - 3x\]`

Now, we can use the fact that AC is equal to the sum of BC and CD:

`\[AC = BC + CD\]`

`\[11x - 290 = 3x - 81 + 196 - 3x\]`

`\[11x - 290 = 115\]`

Solving for x:

`\[11x = 405\]`

`\[x = 37.5\]`

Finally, we substitute this value back into the expression for AC:

`\[AC = 11(37.5) - 290 = 68\]`

Therefore, the length of AC is 68 units. The explanation demonstrates a clear step-by-step approach to solving the problem, from establishing relationships between the given segments to the final calculation of the length of AC using the determined value of x.