Moving from left to right, the points have the following coordinates:

(negative 3 comma 4)

(4 comma negative 3)

(5 comma 3)

Question

What is the area, in square units, of the triangle formed by connecting the three points shown?

Accepted Answer

The correct answer is 24.5 . It's given that a triangle is formed by connecting the three points shown, which are - 3 , 4 , 5 , 3 , and 4 , - 3 . Let this triangle be triangle A. The area of triangle A can be found by calculating the area of the rectangle that circumscribes it and subtracting the areas of the three triangles that are inside the rectangle but outside triangle A. The rectangle formed by the points - 3 , 4 , 5 , 4 , 5 , - 3 , and - 3 , - 3 circumscribes triangle A. The width, in units, of this rectangle can be found by calculating the distance between the points 5 , 4 and 5 , - 3 . This distance is 4 - - 3 , or 7 . The length, in units, of this rectangle can be found by calculating the distance between the points 5 , 4 and - 3 , 4 . This distance is 5 - - 3 , or 8 . It follows that the area, in square units, of the rectangle is 7 8 , or 56 . One of the triangles that lies inside the rectangle but outside triangle A is formed by the points - 3 , 4 , 5 , 4 , and 5 , 3 . The length, in units, of a base of this triangle can be found by calculating the distance between the points 5 , 4 and 5 , 3 . This distance is 4 - 3 , or 1 . The corresponding height, in units, of this triangle can be found by calculating the distance between the points 5 , 4 and - 3 , 4 . This distance is 5 - - 3 , or 8 . It follows that the area, in square units, of this triangle is 1 2 8 1 , or 4 . A second triangle that lies inside the rectangle but outside triangle A is formed by the points 4 , - 3 , 5 , 3 , and 5 , - 3 . The length, in units, of a base of this triangle can be found by calculating the distance between the points 5 , 3 and 5 , - 3 . This distance is 3 - - 3 , or 6 . The corresponding height, in units, of this triangle can be found by calculating the distance between the points 5 , - 3 and 4 , - 3 . This distance is 5 - 4 , or 1 . It follows that the area, in square units, of this triangle is 1 2 1 6 , or 3 . The third triangle that lies inside the rectangle but outside triangle A is formed by the points - 3 , 4 , - 3 , - 3 , and 4 , - 3 . The length, in units, of a base of this triangle can be found by calculating the distance between the points 4 , - 3 and - 3 , - 3 . This distance is 4 - - 3 , or 7 . The corresponding height, in units, of this triangle can be found by calculating the distance between the points - 3 , 4 and - 3 , - 3 . This distance is 4 - - 3 , or 7 . It follows that the area, in square units, of this triangle is 1 2 7 7 , or 24.5 . Thus, the area, in square units, of the triangle formed by connecting the three points shown is 56 - 4 - 3 - 24.5 , or 24.5 . Note that 24.5 and 49/2 are examples of ways to enter a correct answer.

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