GRE_Math_Bible_eBook

204GRE Math Bible

In Case I, θ < 45˚. Whereas, in Case II, θ > 45˚. This is a double case, and the answer therefore is (E).

4.In isosceles triangle ABC, CA = CB = 4. Which one of the following is the area of triangle ABC ?

(A)7

(B)8

(C)9

(D)10

(E)It cannot be determined from the information given

There are many possible drawings for the triangle, two of which are listed below:

206 GRE Math Bible

(2, 3)

(–3, 1)

(0,0)

(4, –2)

(–4, –4)

Example: If the point (a, b) is in Quadrant II and |a| – |b| > 0, then which one of the following is true?

(A)a > 0

(B)b < 0

(C)a > b

(D)|a| < |b|

(E)a + b < 0

We are given that the point (a, b) is in Quadrant II. In Quadrant II, the x-coordinate is negative and the y-coordinate is positive. Hence, a < 0 and b > 0. Reject choices (A) and (B), since they conflict with these inequalities.

Next, since a is negative and b is positive, a < b. Reject Choice (C) since it conflicts with this inequality.

Adding |b| to both sides of the given equation |a| – |b| > 0 yields |a| > |b|. Reject Choice (D) since it conflicts with this inequality.

Since a is negative, |a| equals –a; and since b is positive, |b| equals b. So, a + b = –|a| + |b| = –(|a| – |b|). Now, since |a| – |b| is given to be positive, –(|a| – |b|) must be negative. So, a + b < 0. The answer is (E).

Distance Formula:

The distance formula is derived by using the Pythagorean theorem. Notice in the figure below that the distance between the points (x, y) and (a, b) is the hypotenuse of a right triangle. The difference y – b is the measure of the height of the triangle, and the difference x – a is the length of base of the triangle. Applying the Pythagorean theorem yields

d2 = ( x − a)2 + ( y − b)2

Taking the square root of both sides of this equation yields

d = (x −a)2 + (y − b)2

Coordinate Geometry 207

Example: If C is the midpoint of the points A(–3, –4), and B(–5, 6), then AC =

(A)5

(B)26

(C)61

(D)8

(E)10

Using the distance formula to calculate the distance between A and B yields

AB = (−3− (−5))2 + (−4 − 6)2 = 22 + (−10)2 = 4 +100 = 104 = 226

Midpoint Formula:

The midpoint M between points (x, y) and (a, b) is given by

In other words, to find the midpoint, simply average the corresponding coordinates of the two points.

Example: If (–3, –5) is the midpoint of the part of the line between the x and y axes, then what is the slope of the line?

(A)–5/3

(B)–3/5

(C)3/5

(D)5/3

(E)10

We have that (–3, –5) is the midpoint of the line between the x-intercept (X, 0) and the y-intercept (0, Y). The formula for the midpoint of two different points (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2). Hence, the midpoint of (X, 0) and (0, Y) is ((X + 0)/2, (0 + Y)/2) = (X/2, Y/2). Equating this to the given midpoint yields (X/2, Y/2) = (–3, –5). Equating corresponding coordinates yields X/2 = –3, or X = –6, and Y/2 = –5, or Y = –10. Hence, the slope of the line between (X, 0), which equals (–6, 0), and (0, Y), which equals (0, –10), is

y1 − y2 = x1 − x2

−10− 0 0− (−6) =

−10 =

6 − 53

The answer is (A).

208GRE Math Bible

Slope Formula:

The slope of a line measures the inclination of the line. By definition, it is the ratio of the vertical change to the horizontal change (see figure below). The vertical change is called the rise, and the horizontal change is called the run. Thus, the slope is the rise over the run.

(x, y)

y – b

(a, b)

x – a

Forming the rise over the run in the above figure yields

m= y − b x − a

y – b = m(x – a)

Now, if the line passes through the y-axis at (0, b), then the equation becomes

y – b = m(x – 0)

or

y – b = mx

or

y = mx + b

This is called the slope-intercept form of the equation of a line, where m is the slope and b is the y-intercept. This form is convenient because it displays the two most important bits of information about a line: its slope and its y-intercept.

Coordinate Geometry 209

(A).

Intercepts:

The x-intercept is the point where the line crosses the x-axis. It is found by setting y = 0 and solving the resulting equation. The y-intercept is the point where the line crosses the y-axis. It is found by setting x = 0 and solving the resulting equation.

y

y-intercept

x-intercept x

Example: Graph the equation x – 2y = 4.

Solution: To find the x-intercept, set y = 0. This yields x − 2 0 = 4, or x = 4. So the x-intercept is (4, 0). To find the y-intercept, set x = 0. This yields 0 – 2y = 4, or y = –2. So the y-intercept is (0, –2). Plotting these two points and connecting them with a straight line yields

210GRE Math Bible

Areas and Perimeters:

Often, you will be given a geometric figure drawn on a coordinate system and will be asked to find its area or perimeter. In these problems, use the properties of the coordinate system to deduce the dimensions of the figure and then calculate the area or perimeter. For complicated figures, you may need to divide the figure into simpler forms, such as squares and triangles. A couple examples will illustrate:

Point A has coordinates (0, 4), point B has coordinates (3, 0), and point C has coordinates (5, 1). Using the distance formula to calculate the distances between points A and B, A and C, and B and C yields

AB = (0− 3)2 + (4 − 0)2 = 9+16 = 25 = 5

AC = (0− 5)2 + (4 −1)2 = 25+ 9 = 34

BC = (5− 3)2 + (1− 0)2 = 4 +1 = 5

Adding these lengths gives the perimeter of Triangle ABC:

AB + AC + BC = 5+ 34 + 5

The answer is (A).