Question

Given that AD is the perpendicular bisector of BC , AB=2a+7 , and AC=6a−21 , identify AC . The figure shows triangle A B C with perpendicular bisector A D. AC = 7 AC = 30 AC = 21 AC = 16

Answer 1

So, the correct answer is
`\( AC = 21 \`).

In the given triangle ABC, with AD being the perpendicular bisector of BC, we can apply the Perpendicular Bisector Theorem. According to this theorem, the perpendicular bisector of a segment in a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

Let's denote BD as x and CD as y. Therefore, according to the theorem:

`\[ (AB)/(AC) = (BD)/(CD) \]\\`

Given that
`\(AB = 2a + 7\) and \(AC = 6a - 21\)\\`, we can substitute these values into the equation:

\[ \frac{2a + 7}{6a - 21} = \frac{x}{y} \]

Now, let's find the values of x and y in terms of a. Cross-multiply to get rid of the fractions:

`\[ (2a + 7)y = x(6a - 21) \]`

Now, as AD is the perpendicular bisector, BD = CD. Therefore,
`\(x = y\),`and we can simplify the equation:

`\[ (2a + 7)y = y(6a - 21) \]`

Cancel out the common factor of y:

`\[ 2a + 7 = 6a - 21 \]`

Solve for a:

`\[ 4a = 28 \]`

`\[ a = 7 \]`

Now that we have the value of a, we can find AC:

`\[ AC = 6a - 21 \]`

`\[ AC = 6(7) - 21 \]`

`\[ AC = 42 - 21 \]`

`\[ AC = 21 \]`