# GRE® Coordinate Geometry

#### Concepts Tested from Coordinate Geometry in the GRE

Length of a line segment joining two points, coordinates of a point P that lies on a line segment and divides the line segment in a given ratio, computing the slope of a line if the coordinates of two points through which the line passes is known, testing for collinearity of three or more points, finding the equation of a line and that of parallel and perpendicular lines, computing coordinates of centroid, orthocenter and circumcenter of a triangle, finding equations of median, perpendicular bisector, altitude and angle bisector of a triangle coordinates of whose vertices are known, finding radius of a circumscribing circle, equations of circles, and distance between a point and a line.

1. A straight line 4x + 3y = 24 forms a triangle with the coordinate axes. What is the distance between the orthocentre of the triangle and the centre of the circle that circumscribes the triangle?

1. 10 units
2. 5 units
3. 13 units
4. 12 units
5. 9 units
Approach to solve this Coordinate Geometry Question

Step 1: Compute the coordinates of the Triangle.
Step 2: Determine the coordinates of the orthocentre.
Step 3: Determine the coordinates of the circumcentre.
Step 4: Compute the length of the line segment.

2. Quantitative Comparison

Quantity A Quantity B
Length of the segment of the line 4x + 3y = 12 intercepted between the coordinate axes. Length of the median to side BC of triangle whose coordinates are A(4, 4), B(10, 4) and C(4, 12)

1. Quantity A is greater
2. Quantity B is greater
3. The two quantities are equal
4. Cannot be determined
Approach to solve this GRE Quantitative Comparison Question

Step 1: Quantity A: Compute the length of the line segment joining the x and y intercepts of 4x + 3y = 12.
Step 2: Quantity B: Compute the coordinates of the mid point D.
Step 3: Quantity B: Compute the length of the median AD.
Step 4: The Comparison: Compare the length of the line segment joining x and y intercepts of 4x + 3y = 12 and the median AD of the triangle given in Quantity B.

3. Points C and D trisect the line segment joining points A(4, 5) and B (16, 14). What is the length of the line segment CD?

1. 15 units
2. 5 units
3. 10 units
4. 6 units
5. 3 units
Approach to solve this Coordinate Geometry Question

Step 1: Compute the length of the line segment AB using the distance formula.
Step 2: Because C and D trisect AB, AC = CD = DB.
Step 3: Hence, 1/3rd length of AB = CD.

4. GRE Numeric Entry Question

What is the slope of a line that makes an intercept of 5 in the positive direction of y axis and 1 in the negative direction of x axis?

Approach to solve this GRE Numeric Entry Question

Step 1: Slope of a line = -(y-intercept)/(x-intercept).
Step 2: Substitute values given in the question and compute slope.

5. Which of the following could be the slope of the line that passes through the point (4, 5) and intercepts the y-axis below the origin?
Indicate all that apply.

1. -$$frac{5}{4}$ 2. $\frac{5}{4}$ 3. $\frac{7}{4}$ 4. $\frac{6}{5}$ 5. $\frac{4}{5}$ 6. 2 Approach to solve this Coordinate Geometry Question Step 1: Slope of a line given two points$x1, y1) and (x2, y2) through which the line passes = (y2 - y1)/(x2 - x1).
Step 2: The line passes through (4, 5) and (0, y1) where y1 is the y-intercept of the line.
Step 3: It is given that y1 is negative. Therefore, (5 - y1) will be greater than 5.
Step 4: Compute the range of values that the slope can take and identify answer options that satisfy the condition.

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