# Which of the following points lie within the triangle with vertices (2, 2), (4, -2) and (-2, -2): (1, 2), (2, 1), (3, 2) and (-2, -1). Can this be determined without drawing a graph.

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The vertices of the triangle are (2, 2), (4, -2) and (-2, -2).

First, identify the equation of the lines that make the triangle. The equation of the line joining (2, 2) and (4, -2) is `(y +2)/(x - 4) = (2+2)/(2 - 4)` `=> (y + 2)/(x - 4) = -2` ` => y + 2 = -2x + 8` ` => y = -2x + 6`

The equation of the line joining (4, -2) and (-2, -2) is y = -2. And the equation of the line joining (2, 2) and (-2, -2) is y = x.

y = -2 is the equation of a horizontal line, y = x is the equation of a line with positive slope and y = -2x + 6 is the equation of a line with negative slope.

The points in the triangle lie above the line y = -2, to the right of y = x and to the left of y = -2x + 6. This gives 3 inequalities that have to be satisfied by the points in the triangle: `y >= -2` , `x - y >= 0` and `y + 2x - 6 <= 0`

Consider the four points that are given:

(1, 2) : Here x - y = 1 - 2 = -1 < 0, this point does not lie in the triangle.

(2, 1): Here, y = 1 `>=` -2, x - y = 1 `>=` 0 and y + 2x - 6 = 1 + 4 - 6 = -1 `<=` 0. This point lies in the triangle.

(3, 2): Here y= 2 `>=` -2, x - y = 3 - 2 = 1 `>=` 0 but y + 2x - 6 = 2 + 6 - 6 = 2 > 0. This point does not lie in the triangle.

(-2, -1): Here y = -1 `>=` -2, but x - y = -2 + 1 = -1 < 0. This point does not lie in the triangle.

For conformation, here is the graph of the lines that create the triangle. The points that lie within the triangle and those that do not can be seen.