Decide whether each statement is always true, sometimes true, or never true. Justify your answers. a) Two line segments with the same midpoint have the same length. b) Two parallel line segments have the same midpoint. c) The midpoint of a line segment is the only point that divides it into two equal parts. d) A point equidistant from the endpoints of a line segment is the midpoint of the line segment.

Determine whether each statement is always, sometimes, or never true. Explain. Points A, B, and C determine a plane

Evaluate the following integrals. $\int \frac { z + 1 } { z \left( z ^ { 2 } + 4 \right) } d z$

Sketch the described figure.

Two line segments that both have a midpoint at point $P$ at point $M$

Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. If two rays share a common endpoint, then they form a line.

Why do you think the seafarer tells about his life and its hardships? Cite details from the poem to support your opinion.

The leaves of a tree lose water to the atmosphere via the process of transpiration. A particular tree loses water at the rate of $3 \times 10^{-8} \mathrm{m}^{3} / \mathrm{s}$; this water is replenished by the upward flow of sap through vessels in the trunk. This tree's trunk contains about 2000 vessels, each $100 \mu \mathrm{m}$ in diameter. What is the speed of the sap flowing in the vessels?

In three dimensions, the location of a point can be represented by the ordered triple (x, y, z). a) Find the coordinates of the midpoint of the line segment with endpoints A(2, 3, 1) and B(6, 7, 5). b) Write an expression for the coordinates of the midpoint of the line segment with endpoints $\left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ and $\left( x _ { 2 } , y _ { 2 } , z _ { 2 } \right)$

Determine if each statement is sometimes, always, or never true. Explain your reasoning or provide a counterexample. Through any three points, there is exactly one plane.

Determine whether the following statement is always true, sometimes true, or never true. Justify your answer. Deductive reasoning based on true statements leads to a correct conclusion.

Determine whether each statement is sometimes, always, or never true. Justify your reasoning using a counterexample or proof. The angle bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

Write as powers of 3: $\frac{1}{9}$

Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. $2^{x+1}=5^{1-2 x}$

A charity is organizing a fundraising run along a straight section of highway. On the grid of a roadmap, the starting point is at (23.6, 38.0) and the finish line is at (79.4, 43.8). The charity wants to set up a checkpoint table with water for the runners at the halfway point. Find the coordinates of this checkpoint.