I drew a graph for this, but can't find anything particularly interesting. Is there anything remarkable about these two triangles?

Question

575.Let A = (0,0), B = (4,−3), C = (6,3), P = (−2,7), Q = (9,5), and R = (7,19). Measure the angles of triangles ABC and PQR. Calculate the lengths of the sides of these triangles. Find justification for any conclusions you make.

Answer 1

If you calculate the lengths of the sides of triangle ABC and also for the sides of triangle PQR, you will find that corresponding sides are in proportion. You were asked to measure the angles of each triangle (assume using a protractor), the idea being to discover that corresponding angles are equal. You only need two pair of congruent angles to establish the fact that the triangles are similar (AA similarity). That means the corresponding sides are in proportion.

You can use trigonometry and the Law of Cosines to find the angle measures, but assume you have not learned that yet.

Here is a Desmos graph that calculates the lengths of each side of both triangles, then shows that corresponding sides have the same ratio between them.

desmos.com/calculator/yjiubzf4ek