Exercise 1 Page 352

We are given the following diagram and asked to find the length of $\overline{A B} .$ Let's consider the given diagram.

As we can see, $5 x - 11$ represents the length of $\overline{A B} .$ Therefore, to find $A B,$ we need first to calculate the value of $x .$

From the diagram, we see that $A C$ intersects $\overline{B D}$ at the midpoint $C .$ Moreover, it is perpendicular to $\overline{B D} .$ Thereby, $A C$ is a perpendicular bisector of $\overline{B D} .$ Let's use the Perpendicular Bisector Theorem.

Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

By this theorem,

A

is equidistant from the endpoints of

\overline{B D} .

This means that

\overline{A B}

and

\overline{A D}

have the same measures. From the diagram, we know that the length of

\overline{A B}

is

5 x - 11

and the length of

\overline{A D}

is

3 x + 5 .

Let's set these expressions equal.

5 x - 11 = 3 x + 5

Now, we can solve this equation for

x .

5 x - 11 = 3 x + 5

▼

Solve for

x

$LHS - 3 x = RHS - 3 x$

2 x - 11 = 5

$LHS + 11 = RHS + 11$

2 x = 16

$LHS / 2 = RHS / 2$

x = 8

Now that we know the value of

x,

we can evaluate the expression

5 x - 11

and find the length of

\overline{A B} .

A B = 5 x - 11

$x = 8$

A B = 5 (8) - 11

A B = 40 - 11

Subtract term

A B = 29

The length of

\overline{A B}

is

29 .