Among all of the concepts explored in GMAT, triangles are a commonly revisited concept. When exploring triangles on the coordinate plane, it is important to understand the following terminology:
• Ordered Pair: identification of a point through its coordinates, and is typically written in the form of (a, b) where a is a point on the x-axis, and b is a point on the y-axis
• Origin: point of intersection of x-axis and y-axis, (0, 0)
• Quadrants: a sector on the graph; there are 4 quadrants with coordinates of (+, +) for quadrant I, (-, +) for quadrant II, (-, -) for quadrant III, and (+, -) for quadrant IV
• Slope: steepness of a line defined by the rise over run which is the units that are present vertically divided by the units present horizontally
Upon understanding the terminology, it is crucial to understand the basic equations regarding the distance between two points, or a line, in order to compose a more complex view on triangles.
The basic equation will be y = mx + b where y is a point on the y-axis, m is the slope, x is a point on the x-axis, and b is the intercept on the y-axis.
Since the GMAT normally provides a coordinate graph, counting the units on the y-axis, and the units on the x-axis will be relatively easy.
For example, if two points on the graph have coordinates (0, 0) and (3, 6), we can find the slope by plugging the coordinates into the theorem of rise over run. The difference between the two points will be a value of 3 for the x-axis, and a value of 6 for the y-axis. The slope will be 6/3, which gives a value of 2. Finding the slope is crucial in determining many factors concerning a triangle, and will be often used to determine the coordinates of the three points.
The most basic equation regarding calculating the length of the sides of the triangle will include a^2 + b ^2 = c ^2 where a and b are the two shorter sides, and c is the longer side. This equation is only viable if the triangle is a right-angled triangle. This means that two of the lines are perpendicular from one another, and form a 90 degree angle.
For example, if we are given three points A at (0, 0), B at (3, 0) and C at (3, 5), we can find the length of each side of the triangle easily.
First of all, the line AB would a horizontal line, as there is no difference on the y-axis; thus, by only taking a look at the coordinates on the x-axis, we can determine its length. 3 – 0 = 3.
The line BC would be a vertical line as there is no difference in the x-axis. Similarly, we can find the length by just looking at the coordinates of the y-axis. 5 – 0 = 5. From there, we can use the equation a^2 + b^2 = c ^2, and plug in the values to get.
(3^2) + (5^2) = c^2
9 + 25 = c^2
34 = c^2
c = sqrt(34)
c is approximately 5.83
It is important to write the units out as well. With this information, it will then be easy to determine the perimeter of the triangle. The perimeter will be the sum of all sides. Multiplying the two sides that are perpendicular to one another, and dividing that number by 2 can find the area of the triangle.
In the scenario above, the perimeter of the triangle would be 3 + 5 + 5.83, which would be approximately 13.83. The area would be 5 x 3 / 2, which would be 15.
It is important to note that most of the questions regarding the triangles on the coordinate plane will generally involve right-angled triangles.
Key Takeaway: Test-makers generally attempt to complicate the situation by providing two points, and asking candidates to find the third coordinate capable of constructing a triangle that will meet requirements.
Questions regarding isosceles triangles and any other types of triangles will require different equations; however, these are generally not found in coordinate plane questions, as it is difficult for candidates to find the angles involved.
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