Find the slope of a line perpendicular to g(t)

I) g(t) passes through the points (9,6) and (4,-13)

II) g(t) does not pass through the origin 

Perpendicular lines have opposite reciprocal slopes. For instance: a line with a slope of   would be perpendicular to a line with slope of .

To find the slope of a line, we just need two points.

I) Gives us two points on g(t). We could find the slope of g(t) and then the slope of any line perpendicular to g(t).

So the slope of a line perpendicular to g(t) is equal to:

II) Is irrelevant or at least not helpful.

We are given a line, f(x), and asked to find the slope of another line, h(x).

I) Gives a point on h(x). We could plug in the point and solve for our slope. When we do this since x=0 we are unable to find the value for our slope. Therefore, statement I is not sufficient to solve the question.

II) Tells us the two lines are perpendicular. Take the opposite reciprocal of the slope of f(x) to find the slope of h(x).

Therefore,

and thus the slope of h(x) will be,

.

Statement II is sufficient to answer the question. 

To find the equation of any line, we need 2 pieces of information, the slope of the line and any point on the line. From statement 1, we get a point on Line AB. From statement 2, we get the slope of Line BC. Since we know that AB is perpindicular to BC, we can derive the slope of AB from the slope of BC. Therefore to find the equation of the line, we need the information from both statements. 

To find the equation of a perpendicular line you need the slope of the line and a point on the line. We can find the slope by knowing g(x).

I) Gives us a point on h(x).

II) Gives us the y-intercept of h(x).

Either of these will be sufficient to find the rest of our equation.

Statement 1: We're given the equation to line AB which contains the slope. Because the line we're being asked for is perpendicular to it, we know the slope will be its inverse.

The slope of our line is then 

Statement 2: We can write the equation to the perpendicular line only if we have a point that falls within that line. Luckily, we're given such a point in statement 2.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Find the equation of the line perpendicular to r(x)

I) r(x) has a slope of -15

II) The line must pass through the point (9, 96)

Recall that perpendicular lines have opposite reciprocal slopes.

Use I) to find the slope of our new line

Use II) along with our slope to find the y-intercept of our new line.

Therefore both statements are needed.

Recall that perpendicular lines have opposite, reciprocal slopes. We can find the slope of  from the question.

Statement I gives us a point on , which we can use to find the y-intercept of , and then the equation.

The slope of  must be the opposite reciprocal of , this makes our slope .

Statement I tells us that  passes through the point , so we can use slope-intercept form to find our equation:

So, our equation is

Statement II gives us a point on , which does not help us in the slightest with . Therefore, only Statement I is sufficient.

Assume Statement 1 alone. The -intercept of the line of the equation can be found by substituting  and solving for :

The -intercept of the line is the origin ; it follows that this is also the -intercept. 

Therefore, Statement 1 alone yields only one point of the line, from which its equation cannot be determined.

Assume Statement 2 alone. The slope of the line of the equation  can be calculated by putting it in slope-intercept form :

The slope of this line is the coefficient of , which is . A line perpendicular to this one has as its slope the opposite of the reciprocal of , which is 

.

However, there are infintely many lines with this slope, so no further information can be determined.

Now assume both statements to be true. From Statement 1, the slope of the line is , and from Statement 2, the -coordinate of the -initercept is . Substitute in the slope-intercept form:

Statement 1: Since the line we're looking for is perpendicular to line XY, our slope will be the inverse of line XY's slope .

The slope of our line is then . Just knowing the slope however, is not sufficient information to answer the question.

Statement 2: We're provided with a point which will allow us the write the equation.

Since the two equations are in slope-intercept form,  coefficients  and  are the slopes of the two lines.

If  , then this tells us that one of slopes  and  is positive and one is negative; this only eliminates the possibility of the lines being parallel.

If  - or, equivalently, , then each of the slopes  and  is the opposite of the reciprocal of the other. This makes the lines perpendicular.