Question

Given that AD is the perpendicular bisector of BC, AB=2a+7, and AC=6a−21, identify AC.

Answer 1

Since AD is the perpendicular bisector of BC, AB and AC must be equal in length. By setting the given expressions for AB and AC equal to each other and solving for 'a', we find that AC equals 21 units.

Step-by-step explanation:

Since AD is the perpendicular bisector of BC in a triangle, it means that AB and AC are equal in length because the perpendicular bisector of a line segment not only bisects the line segment into two equal parts but also shows that any point on the perpendicular bisector is equidistant from the endpoints of the line segment it bisects. Therefore, we can equate AB and AC because of this property.

AB = AC

Given that AB = 2a + 7 and AC = 6a - 21, we can set these two expressions equal to each other because AB and AC must be the same length:

2a + 7 = 6a - 21

Next, we rearrange the equation to solve for a:

2a - 6a = -21 - 7

-4a = -28

a = 7

Now, we can find AC by substituting a back into one of the original equations:

AC = 6a - 21

AC = 6(7) - 21

AC = 42 - 21

AC = 21

Therefore, the length of AC is 21 units.

Answer 2

Since AD is the perpendicular bisector of BC, ABC is an isosceles triangle with AB = AC. Solving 2a+7 = 6a-21 for 'a' gives us AC = 21 units.

Step-by-step explanation:

If AD is the perpendicular bisector of BC, this means that triangle ABC is isosceles with AB = AC. Thus, if AB is given as 2a+7, then AC must be equal in length to AB. Hence, AC also measures 2a+7. Given that AC is originally stated to be 6a-21, we set 2a+7 equal to 6a-21 to solve for 'a'. After finding the value of 'a', we substitute it back into the expression for AC to find the specific length of AC.

Here's the calculation:

2a + 7 = 6a - 21

4a = 28

a = 7

AC = 2a + 7

AC = 2(7) + 7

AC = 21

Hence, the length of AC is 21 units.