I can’t remember where I first saw the coordinate geometry – triangles tasks that we use. I think the first one came from some NCTM publication.
The vertices of the triangle PQR are the points P(1,2), Q(4,6) and R(–4,12). Which one of the following statements about triangle PQR must be true?
a. PQR is a right triangle with the right angle at P.
b. PQR is a right triangle with the right angle at Q.
c. PQR is a right triangle with the right angle at R.
d. PQR is not a right triangle.
Justify your answer.
How would you justify whether or the triangle is right?
My first inclination is to show that the product of the slopes of lines RQ and PQ is –1. Most of my students did this as well. Is that what you would do?
I’m always glad for a student who suggests that we could use distances to show that RQ2+PQ2=RP2 … just another reminder that we don’t all have to have the same justification to arrive at the same, correct solution.
The next task is about the centroid of the triangle.
A triangle has vertices (4,7), (7,9), and (10,6). What are the coordinates of the centroid of the triangle?
Use your results to describe how to calculate the coordinates of the centroid given three vertices of a triangle (a,b), (c,d), and (e,f).
How would your students determine the coordinate of the centroid? We provided our students an opportunity to look for regularity in repeated reasoning using dynamic geometry software. What do you notice? Eventually I will give you three vertices of a triangle and ask you for the coordinates of the centroid when you don’t have access to technology. Can you generalize your results? I watched students play using Class Capture. I love the evidence of the calculations on the screens with the Scratchpad showing that students were making sense of what they saw on the Graphs page and were beginning to generalize their results.
And then before we generalized the results as a class, I sent a Quick Poll to assess what students had learned.
Almost everyone was successful working in groups. Except that looking at the question in the Review Workspace – graph, I realized that the three “vertices” I gave were collinear. Oops.
CCSS-M G-CO.C.10 says Prove theorems about triangles. Theorems include: … the medians of a triangle meet at a point.
Do you use coordinate geometry to prove the medians of a triangle meet at a point? Maybe eventually we will try that proof.
Another problem in this lesson that I like is the following:
The midpoints of a triangle are (2,6), (5,11), and (10,8). What are the vertices of the triangle?
Use yours results to describe how to find the vertices of the triangle given three midpoints of a triangle (a,b), (c,d), and (e,f).
How would you determine the vertices of the triangle? I didn’t get to see my students work on this one, as they did it while I was at a conference, but I look forward to seeing their work when we return to school tomorrow.
And one more: Find the point of intersection of the angle bisectors of the interior angles of a triangle whose vertices are (–2,–3), (32,–17), (1/2,29/2).
And so the journey continues … even when I make up impossible problems.