# GRE® Quantitative Comparison

Comparing length of line segments. Coordinate geometry

This GRE quant practice question is a quantitative comparison question in coordinate geometry. Concept: Finding length of two line segments based on different data available abouts these segments and comparing which of the two lines is longer.

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. The relationship cannot be determined from the information given

Video Explanation will be added soon

### Approach to the answer

Use these hints to get the answer

1. Find x intercept and y intercept of the line 4x + 3y = 12.
2. Find the length of the segment of the line 4x + 3y = 12 intercepted between the axes. The end points of the line segment are the x and y intercepts of the line.
3. For line 2, median to side BC will meet side BC at its midpoint.
4. Find the coordinates of the midpoint of the line segment BC.
5. Find the length of the median - the length of the segment from vertex A to the midpoint of BC.

### Evaluate Quantity A

Find the x intercept and y intercept of the line 4x + 3y = 12
The end points of the segment of the line intercepted between the axes will be the x intercept and intercept of the line.

#### Compute the x intercept of the line

The y coordinate of the point at which the line intercepts the x axis is 0.
Therefore, to compute the x-intercept of a line, substitute the value of y as 0 in the equation of the line.
4x + 3(0) = 12
4x = 12
Or x = 3. The x-intercept of the line is 3.

#### Compute the y intercept of the line

The x coordinate of the point at which the line intercepts the y axis is 0.
Therefore, to compute the y-intercept of a line, substitute the value of x as 0 in the equation of the line.
4(0) + 3y = 12
3y = 12
Or y = 4. The y-intercept of the line is 4.

#### Compute the length of the line segment, coordinates of whose end points are the intercepts

Coordinates of one of the end points of the line segment is (3, 0) and that of the other end point is (0, 4).
Length of a line segment coordinates of whose end points are (x1, y1) and (x2, y2 is $$sqrt {{\left$ {{{x}_{2}-{x}_{1}}} $right$}^{2}+{$left$ {{{y}_{2}-{y}_{1}}} $right$}^{2}}) So, length of the line segment joining points$3, 0) and (0, 4) = $$sqrt {{\left$ {{3-0}} $right$}^{2}+{$left$ {{4-0}} $right$}^{2}}) = $\sqrt {9+16}$ = $\sqrt {25}$ = 5 units. ### Evaluate Quantity B #### Compute the coordinates of the mid point D of side BC The coordinates of the midpoint of a line segment AB, coordinates of whose end points are A$x1, y1) and B(x2, y2) = $frac {{x}_{1}+{{x}_{2}}} {2},$frac {{y}_{1}+{{y}_{2}}} {2}$) The coordinates of the end points of BC are B$10, 4) and C(4, 12).
So, the coordinates of the midpoint D = $frac {10+4} {2},$frac {4+12} {2}$) =$7, 8).

#### Compute the length of the median AD.

Length of a line segment coordinates of whose end points are (x1, y1) and (x2, y2 is $$sqrt {{\left$ {{{x}_{2}-{x}_{1}}} $right$}^{2}+{$left$ {{{y}_{2}-{y}_{1}}} $right$}^{2}}) The coordinates of the end points of median AD are A$4, 4) and D(7, 8).
Therefore, length of median AD = $$sqrt {{\left$ {{7-4}} $right$}^{2}+{$left$ {{8-4}} \right$}^{2}}) = $\sqrt {{3}^{2}+{4}^{2}}$ = 5 units

### The Comparison

Quantity A. Length of the segment of the line 4x + 3y = 12 intercepted between the axes = 5 units
Quantity B. Length of the median to the side BC = 5 units
Both quantities are equal. Choice C is the answer.