Question

Question #b7d4d

Answer 1

`y = 4`

Explanation:

For these problems, you must understand what you are given, and what you are trying to find. This means understanding any definitions and initial information. Once that is done, it is just a matter of identifying what steps can take you from the initial information to the desired conclusion. Let's use this problem as an example.

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Given: Two points `A(1,3)` and `B(1,5)`

Goal: The perpendicular bisector of the line segment `bar(AB)`

To proceed, we must know what a perpendicular bisector is.

The perpendicular bisector of a line segment is a line segment, ray, or line which intersects the initial line segment at its midpoint, forming a `90^@` angle.

With that, we know we want a line which passes through the midpoint of `(1,3)` and `(1,5)`, and has a slope which makes it form a `90^@` angle with `bar(AB)`.

The midpoint of two points `(x_1, y_1)` and `(x_2, y_2)` is given by `((x_1+x_2)/2, (y_1+y_2)/2)`. Applying this to `A` and `B`, we get their midpoint as `((1+1)/2, (3+5)/2) = (1, 4)`

Next, to find the desired slope, we can note that `bar(AB)` is a vertical line segment, meaning our desired line will be horizontal, giving it a slope of `0`. If the slope of `bar(AB)` was not vertical or horizontal, we could also use the property that two lines with slopes `m_1, m_2` are perpendicular if and only if `m_1 = -1/m_2`.

Finally, we can get our line by plugging in the point it passes through and its slope into the point-slope equation of a line: `y-y_1 = m(x-x_1)`. This gives is our desired equation as

`y-4 = 0(x-1)`

`=> y-4 = 0`

`:. y = 4`

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Note that when solving, in addition to the given information, we use definitions, theorems, and other properties not given to help find the solution. The more of these you have at your disposal, the more problems you can solve, and the easier they become.