Find the area of a triangle given the coordinates of its vertices

Question

The following is a GRE practice test question which I don't understand.

According to this general argument, it appears that any arbitrary point $(x, 7)$ would have "height" equal to 10 but then if we have $x = 1000000$ for example it would appear that the area of the triangle would be significantly larger than just 30 units.

Or am I misunderstanding the argument given in this explanation?

Answer 1

According to this general argument, it appears that any arbitrary point $(x,7)$ would have "height" equal to 10 but then if we have $x=1000000$ for example it would appear that the area of the triangle would be significantly larger than just 30 units.

Or am I misunderstanding the argument given in this explanation?

As others have correctly pointed out, the height of a triangle is a measurement perpendicular to the base -- in this case, a measurement strictly in the direction of the y-axis, specifically $7 - (-3) = 7 + 3 = 10$. Indeed, making the upper point any value $(x, 7)$ would maintain the same height and therefore the same area of 30.

Intuitively, I would disagree that it would "appear" that the area of the triangle would be significantly larger if that were adjusted. While the triangle would certainly get longer, it would simultaneously get skinnier and thus the area would balance out to the same value. For example, here's the graph with an upper point at $(1000, 7)$ from Wolfram Alpha:

Side note: If you do enter the point with $x = 1000000$, then Wolfram Alpha chokes and erroneously says "not a possible triangle"! Apparently it really doesn't like a triangle so negligibly skinny.

Answer 2

The height of a triangle is the minimum distance between the base and the opposite vertex (minimum because in a right triangle the perpendicular is the shortest side).You can compare this with measuring the height of a mountain.Do you measure the height in a slanting way?Draw a parallel line to the base through $(28,7)$.So,all points on that line are equidistant from the base.That's the main principle here.

Answer 3

The area of triangle is half base length multiplied by height/altitude.It can also be stated that area is same for all elongated triangles between two parallels with same base length (on one parallel line) and vertex ( on the other parallel line) and all sharing the same altitude... no matter where the apex is placed on the line parallel to the base.

Depending on which point you choose to call the vertex, and perpendicular length/height dropped from it to the opposite side, there are three pairs of (base, altitude) choices for same product area.

Answer 4

Imagine a stack of pennies. You can slide the pennies to make a slanted cylinder |>/ In this transformation the total volume of the pennies didn't change. This is the same principle except in 2-D. Hope that helped!