Question

Let A(2,1) and B(10,7) be points in the xy-plane. Let R be the region in the first quadrant consisting of all points C for which triangle A B C has three acute angles.

Answer 1

The question involves using vector addition to find the resultant displacement R in the context of creating an acute triangle ABC in the xy-plane.

Step-by-step explanation:

The student's question asks about a geometric construction involving vectors and displacement in the context of the xy-plane and acute triangles. Points A and B are positions in the plane, and vector R represents the resultant displacement vector when combining these two points. The resultant R can be obtained geometrically or by using analytical methods such as the Pythagorean theorem and trigonometry.

To ensure that triangle ABC has all acute angles, point C must be placed in such a way that none of the angles in the triangle exceeds 90 degrees. Moreover, in vector terms, the resultant R can be found by the sum of vectors A and B, while the difference, represented as D, is the longer diagonal in the parallelogram formed with A and B.