Identify the distance between the points $$ (2,4,12) $$ and $$ (7,4,0) $$, and identify the midpoint of the segment for which these are the endpoints. Round to the nearest tenth, if necessary. $$ d \approx 17 $$ units; $$ M(4.5,4,6) $$ $$ d \approx 13 $$ units; $$ M(4.5,4,6) $$ $$ d \approx 17 $$ units; $$ M(2.5,0,6) $$ $$ d \approx 13 $$ units; $$ M(2.5,0,6) $$

Question

UpStudy AI · Accepted Answer

We are given the points $$ A(2,4,12) $$ and $$ B(7,4,0) $$.

**Step 1. Calculate the distance $$ d $$:**

The distance formula in three dimensions is  
$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}.
$$

Substitute the coordinates:
$$
d = \sqrt{(7-2)^2 + (4-4)^2 + (0-12)^2} = \sqrt{5^2 + 0^2 + (-12)^2}.
$$

Simplify:
$$
d = \sqrt{25 + 0 + 144} = \sqrt{169} = 13.
$$

**Step 2. Find the midpoint $$ M $$ of segment $$ \overline{AB} $$:**

The midpoint formula in three dimensions is  
$$
M = \left( \frac{x_1+x_2}{2}, \,\frac{y_1+y_2}{2}, \,\frac{z_1+z_2}{2} \right).
$$

Substitute the coordinates:
$$
M = \left( \frac{2+7}{2}, \,\frac{4+4}{2}, \,\frac{12+0}{2} \right) = \left( \frac{9}{2}, \,4, \,6 \right) = (4.5,4,6).
$$

Thus, the distance and midpoint are:
$$
d = 13 \text{ units}, \quad M(4.5,4,6).
$$

The correct choice is:  
$$ d \approx 13 $$ units; $$ M(4.5,4,6) $$.
The distance between the points is approximately 13 units, and the midpoint is at (4.5, 4, 6).

Identify the distance between the points and
, and identify the midpoint of the segment for which
these are the endpoints. Round to the nearest tenth, if
necessary.
units;
units;
units;
units;

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Geometry Questions

Identify the distance between the points and , and identify the midpoint of the segment for which these are the endpoints. Round to the nearest tenth, if necessary. units; units; units; units;