# Engaging students: Defining the term perpendicular bisector

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Biviana Esparza. Her topic, from Geometry: defining the term perpendicular bisector.

A2. How could you as a teacher create an activity or project that involves your topic?

A fun project that involves perpendicular bisectors is a project that I did in my project-based instruction class earlier this semester. The geometry project required students to create a piece of origami that had an angle bisector and a perpendicular bisector labeled. Leading up to the project creation and presentation day was a series of workshops and DIY activities in which students learned what terms such as congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, perpendicular bisector, circumcenter, and incenter. These activities included working with patty paper to create angle bisectors and perpendicular bisectors on triangles, worksheets where students had to graph triangles and find the circumcenter and incenter, independent practice, formative assessments, and lastly the final origami creation. It was fun to see students take ownership of their learning and be proud of their final origami creation, because they were allowed to create whatever they wanted as long as an angle bisector and a perpendicular bisector were labeled. Students had a firm understanding of what the key vocabulary terms were, especially perpendicular bisectors and angle bisectors, because they had used them so much throughout the workshops.

C1. How has this topic appeared in pop culture?

Paper Planes is an Australian film released in 2015 about a boy named Dylan who has a talent for making paper planes and wants to go the World Paper Plane Championship. In the movie, Dylan is taught how to make the “perfect” paper plane by a student teacher. Students start off by making simple planes like those that most people make. Although most people making paper airplanes don’t think of terms such as perpendicular bisectors or angle bisectors, they are the basics to making any form of paper airplane. The first step to making a plane, which is folding a piece of paper in half by aligning two opposite edges, creates a perpendicular bisector: the fold is a perpendicular bisector to the edges it touches. Students in a class learning about perpendicular bisectors could be shown minutes 5:40 to 8:21 to engage them about paper airplanes and they could be asked how paper planes could be related to geometry. This could show them that something as simple as a paper plane has some mathematical connections.

E1. How can technology be used to effectively engage students with this topic?

Gizmos is a website full of interactive simulations and lesson plans that effectively incorporate technology in the classroom. The website has a lesson titled “Segment and Angle Bisectors” in which students manipulate points to explore the properties of perpendicular bisectors and points on an angle bisector. This is a helpful tool to let students discover properties on their own instead of the teacher directly telling them what a perpendicular bisector is. The website also includes a worksheet with questions to go along with the gizmo exploration.

References

https://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=174

# Area of a triangle: Incenter (Part 6)

The incenter $I$ of a triangle $\triangle ABC$ is defined by the intersection of the angle bisectors of its three angles. A circle can be inscribed within $\triangle ABC$, as shown in the picture.

This incircle provides a different way of finding the area of $\triangle ABC$ commonly needed for high school math contests like the AMC 10/12. Suppose that the sides $a$, $b$, and $c$ are known and the inradius $r$ is also known. Then $\triangle ABI$ is a right triangle with base $c$ and height $r$. So

$\hbox{Area of ~} \triangle ABI = \displaystyle \frac{1}{2} cr$

Similarly,

$\hbox{Area of ~} \triangle ACI = \displaystyle \frac{1}{2} br$

$\hbox{Area of ~} \triangle BCI = \displaystyle \frac{1}{2} ar$

Since the area of $\triangle ABC$ is the sum of the areas of these three smaller triangles, we conclude that

$\hbox{Area of ~} \triangle ABC = \displaystyle \frac{1}{2} r (a+b+c)$,

or

$\hbox{Area of ~} \triangle ABC = rs$,

where $s = (a+b+c)/2$ is the semiperimeter of $\triangle ABC$.

This also permits the computation of $r$ itself. By Heron’s formula, we know that

$\hbox{Area of ~} \triangle ABC = \sqrt{s(s-a)(s-b)(s-c)}$

Equating these two expressions for the area of $\triangle ABC$, we can solve for the inradius $r$:

$r = \displaystyle \sqrt{ \frac{(s-a)(s-b)(s-c)}{s} }$

For much more about the inradius and incircle, I’ll refer to the MathWorld website.