GRE® Quant Practice | Coordinate Geometry Q2

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

Question 2 : Coordinate Geometry 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

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Video Explanation

Play Video: GRE Coordinate Geometry Length of line segment Video Solution

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Explanatory Answer

Evaluate Quantity A : Length of the segment of the line 4x + 3y = 12 intercepted between the coordinate axes.

Compute the x intercept of the line 4x + 3y = 12:

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.
Equation of the line, 4x + 3y = 12
4x + 3(0) = 12 → 4x = 12 or x = 3.
Therefore, the x-intercept of the line is 3.

Compute the y intercept of the line 4x + 3y = 12

Equation of the line, 4x + 3y = 12
4(0) + 3y = 12 → 3y = 12 or y = 4.
Therefore, the y-intercept of the line is 4.

Compute the length of the line segment joining the x and y intercepts of 4x + 3y = 12:

Coordinates of one of the x and y intercept of the line are (3, 0) and (0, 4) respectively.
Length of a line segment, coordinates of whose end points are (x₁, y₁) and (x₂, y₂) is \\sqrt{(x_2 - x_1)^2 + (y_2 - y_2)^2})
So, length of line segment joining points (3, 0) and (0, 4) = \\sqrt{(3 - 0)^2 + (4 - 0)^2}) = \\sqrt{9 + 16}) = \\sqrt{25}) = 5 units

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Evaluate Quantity B : Length of the median to side BC of triangle whose coordinates are A(4, 4), B(10, 4) and C(4, 12)

Compute the coordinates of the mid point D of side BC:

The coordinates of the midpoint of a line segment BC, coordinates of whose end points are B(x₁, y₁) and C(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{(x_2 - x_1)^2 + (y_2 - y_2)^2})
The coordinates of the end points of median AD are A(4, 4) and D(7, 8).
Therefore, length of median AD = \\sqrt{(7 - 4)^2 + (8 - 4)^2}) = \\sqrt{(3)^2 + (4)^2}) = \\sqrt{25}) = 5 units

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The Comparison

Quantity A: Length of the segment of the line 4x + 3y = 12 is 5 units
Quantity B: Length of the median to the side BC is 5 units
Both quantities are equal.

Choice C is the correct answer