Find the exact length of line segment $$ \overline{P Q} $$ given $$ P(x, y)=(8,-17) $$ and the midpoint of $$ \overline{P Q} $$ is $$ M(x, y)=(7,-16) $$. $$ \overline{P Q}= $$ Check Answer

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Let the coordinates of point $$ Q $$ be $$(x_Q, y_Q)$$. Since $$ M $$ is the midpoint of $$ \overline{PQ} $$, we have:

$$
\frac{8 + x_Q}{2} = 7 \quad \text{and} \quad \frac{-17 + y_Q}{2} = -16.
$$

Solving the first equation:

$$
\frac{8 + x_Q}{2} = 7 \implies 8 + x_Q = 14 \implies x_Q = 14 - 8 = 6.
$$

Solving the second equation:

$$
\frac{-17 + y_Q}{2} = -16 \implies -17 + y_Q = -32 \implies y_Q = -32 + 17 = -15.
$$

Thus, point $$ Q $$ is $$(6, -15)$$.

Now, the length of $$ \overline{PQ} $$ is given by the distance formula:

$$
\overline{PQ} = \sqrt{(x_Q - x_P)^2 + (y_Q - y_P)^2} = \sqrt{(6 - 8)^2 + (-15 - (-17))^2}.
$$

Simplify inside the square root:

$$
(6 - 8)^2 = (-2)^2 = 4,
$$
$$
(-15 - (-17))^2 = (2)^2 = 4.
$$

Thus,

$$
\overline{PQ} = \sqrt{4 + 4} = \sqrt{8}.
$$

Finally, simplify $$ \sqrt{8} $$:

$$
\sqrt{8} = 2\sqrt{2}.
$$

Hence, the exact length of $$ \overline{PQ} $$ is $$ 2\sqrt{2} $$.
The exact length of $$ \overline{PQ} $$ is $$ 2\sqrt{2} $$.

Find the exact length of line segment given and the midpoint of
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