Geometry - GMAT Quantitative

To find the slope of a line perpendicular to a given line, simply take the opposite reciprocal of the slope of the given line.

Since f(x) is given in slope intercept form,

.

Therefore our original slope is

So our new slope becomes:

Let one of the angles measure ; then the other angle measures . The sum of the measures of the acute angles of a triangle is , so we can set up and solve this equation:

The acute angles measure ; since we want the smaller of the two, is the correct choice.

The equation for a line in standard form is written as follows:

Where is the slope of the line and is the y intercept. By definition, the slope of a line is the negative reciprocal of the slope of the line to which it is perpendicular. So if the given line has a slope of , the slope of any line perpendicular to it will have the negative reciprocal of that slope. This gives us:

Triangle ABC is an isosceles right triangle. Therefore, its angles at the basis BC will always be .

This stems from the fact that the sums of the angles of a triangle are and in our case with ABC a right and isosceles triangle, , therefore for the two remaining angles are equal.

There are 90 degrees left, therefore to find the measure of each angle we do the following,

.

Here, we can tell the size of the angles by recognizing the length of the sides indicative of a right triangle with angles .

Indeed, the length of the sides are and . Any triangle with sides , where is a constant, will have angles .

In our case . Therefore, angle will be the smallest of the three possible angles, since it is between the two longest sides ( the hypotenuse and AB, which is longer than AC). Therefore the larger angle will be thus arriving at our final answer.

Since here ABC is a isosceles right triangle, its height is half the size of the hypotenuse.

We just need to apply the Pythagorean Theorem to get the length of BC, and divide this length by two.

, so .

Therefore, and the final answer is .

Assume Statement 1 alone. In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. The -intercept of the line of the equation can be found by substituting and solving for :

The -intercept of the line is at the origin, . It follows that the -intercept is also at the origin. Therefore, Statement 1 only gives one point on the line, and its slope 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

.

The question is answered.

The slope of the first line, in terms of , is

The slope of the second line is

The slopes of two perpendicular lines have product , so we set up this equation and solve for :

or

We can eliminate the choice immediately since the slopes of two perpendicular lines cannot have the same sign. We can also eliminate and undefined, , since a line with slope 0 and a line with undefined slope are perpendicular to each other, not a line of slope -1 or 1.

Of the two remaining choices, we check for the choice that includes two numbers whose product is -1.

and

so is the correct choice.

Parallel lines have the same slope. Therefore, the slope of the new line is , as the equation of the original line is ,with slope .

and :

In order for one line to be perpendicular to another, its slope must be the negative reciprocal of that line's slope. That is, the slope of any perpendicular line must be opposite in sign and the inverse of the slope of the line to which it is perpendicular:

In the given line we can see that , so the slope of any line perpendicular to it will be:

There is only one answer choice with this slope, so we know the following line is perpendicular to the line given in the problem:

For a given line defined by the equation with slope , any line perpendicular to has a slope of , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is .

For a given line defined by the equation with slope , any line perpendicular to has a slope of , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is .

For a given line defined by the equation with slope , any line perpendicular to has a slope of , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is .

An angle can be given a name with three letters if the middle letter is the name of the vertex and the other two letters denote points on different sides of the angle. All four of the three-letter choices fit this description.

An angle can be given a one-letter name if the letter is the name of the vertex and if it is the only angle shown in the diagram to have that vertex (thereby avoiding ambiguity). There are four angles in the diagram with vertex , so we cannot use to indicate any of them, including . is the correct choice.

Two angles form a linear pair if they have the same vertex, they share one side, and their interiors do not intersect. has vertex and sides and ; has vertex , shares side , and shares no interior points, so this is the correct choice.

Parallel lines have the same slope. Therefore, we need to find the slope once both equations are in slope intercept form :

The lines are parallel because the slopes are the same.

can be seen to be completely contained in - that is, . The union of a set and its subset is the containing set, so the correct response is .

Lines are perpendicular if their slopes are negative reciprocals of each other. First we need to find the slope of the line in the question stem.

The negative reciprocal of 3 is , so our answer will have a slope of . Let's go through the answer choices and see.

: This line is of the form , where is the slope. The slope is 3, so this line is parallel, not perpendicular, to our line in question.

: The slope here is , also wrong.

: The slope of this line is . This is the reciprocal, but not the negative reciprocal, so this is also incorrect.

The line between the points :.

This is the correct answer! Let's check the last answer choice as well.

The line between points :, which is incorrect.