Points A (5; 3), B (0; 0), C (10; -1) are given on the coordinate plane. Determine the type of angle ABC.

In this task, you need to determine the type of angle abc (straight, obtuse, developed or sharp), if the coordinates of the points are known:

a (5; 3);
in (0; 0);
with (10; – 1).

Plotting the abc angle
We mark points a, b and c on the coordinate plane and draw the angle abc:

For comparison, mark the point d (3; – 5) in this figure.

Obviously, the abc angle is less than the abd angle. Let us prove that angle abd is right.

Selecting equal triangles in the picture
Let us construct perpendiculars aa1 and dd1 to the abscissa and ordinate axes, respectively.

Point a1 has coordinates (5; 0), point d1 has coordinates (0; – 5).

Consider triangles baa1 and bdd1. They are rectangular and have legs of the same length. Therefore, these triangles are equal.

Point b is common for triangles baa1 and bdd1, so we can conclude that triangle baa1 can be obtained by rotating triangle bdd1 by angle abd.

Consequently, ∠avd = ∠а1вd1. The angle a1 to d1 is the angle between the coordinate axes, and thus is equal to 90 °.

Conclusion about the type of angle abc
Since the abc angle is less than the right angle, this angle is acute.

Answer: the abc angle is sharp.