Thinking outside the triangle

In grade 10 we’ve been working on classifying triangles using their coordinate points, and calculating perimeter and area.  We worked on this one yesterday

It was an isosceles triangle with a horizontal side, which made calculating the area pretty easy. We used distance formula to find the side lengths and then added them up to find the perimeter.

We then practiced finding the equation of the line containing the perpendicular bisector (médiatrice) of the non-congruent side…and compared the result to what we got when we calculated the equation of the median through C (the angle between the two congruent sides).

We had to review definitions, and make plans.  A lot of this math is simple to calculate, but if you are not careful you can spend your time calculating unnecessary things.

Today we had 2 more triangles to work with

This was a right angle triangle that is also scalene.  We proved it was a right triangle using slopes, and also using the pythagorean theorem.  Once we know it is a right angle triangle we can use the perpendicular sides as base and height, and calculate the area.

The last triangle was not a right angle triangle.

We needed to calculate the area.  To do so we needed to calculate the height (altitude) of the triangle (the shortest distance between a point and the base).  We created equations, substituted and solved.  It took a long time to go through the process.

Afterwards we looked outside the triangle, and saw an elegant approach to finding the area.

We can extend lines vertically and horizontally to create a rectangle enclosing the triangle.  We can calculate the area of the rectangle, and then subtract the areas of the right angle triangles around our interior triangle.  It’s so neat to see how these problems can be solved with many different approaches.