6.2: Hanging with the Incenter
This activity is intended to supplement Geometry, Chapter 5, Lesson 3.
Problem 1 – Exploring the Angle Bisector Theorem
Start the Cabri Jr. application by pressing APPS and selecting CabriJr. Open the file ANGBIS by pressing \begin{align*}Y=\end{align*}, selecting Open..., and selecting the file.
Line \begin{align*}BD\end{align*} is the angle bisector of \begin{align*}\angle{ABC}\end{align*}. Find \begin{align*}DE\end{align*} and \begin{align*}DF\end{align*} using the Distance and Length tool (press GRAPH and select Measure > D.&Length). Remember that \begin{align*}DE\end{align*} means “the length of \begin{align*}\overline{DE}\end{align*}.”
1. Move point \begin{align*}D\end{align*} to 4 different positions and record the measurements in the table below. To move the point, move the cursor over the point, press ALPHA, move the point to the desired location, then press ALPHA again to release the point.
Position | \begin{align*}1^{st}\end{align*} position | \begin{align*}2^{nd}\end{align*} position | \begin{align*}3^{rd}\end{align*} position | \begin{align*}4^{th}\end{align*} position |
---|---|---|---|---|
\begin{align*}DE\end{align*} | ||||
\begin{align*}DF\end{align*} |
2. What is the relationship between the measurements of \begin{align*}DE\end{align*} and \begin{align*}DF\end{align*}?
3. Complete the following statement: If a point is on the bisector of an angle, then the point is _____________________ from the sides of the angle.
Problem 2 – Exploring the Incenter of a Triangle
Open a new Cabri Jr. file by pressing \begin{align*}Y=\end{align*}, selecting New, and answer no if asked to save. Construct an acute \begin{align*}\triangle{ABC}\end{align*} and construct the angle bisector of all three angles. Using Cabri Jr., answer the following questions.
4. What do you notice about the angle bisectors of all three angles?
5. The point of concurrency for the angle bisectors is the incenter. Create and label this point \begin{align*}R\end{align*}. Can you move vertex \begin{align*}A\end{align*} so that the incenter is on a side of \begin{align*}\triangle{ABC}\end{align*}? If so, what kind of triangle is \begin{align*}ABC\end{align*} in this case?
6. Can you move vertex \begin{align*}A\end{align*} so that the incenter is outside of \begin{align*}\triangle{ABC}\end{align*}? If so, what kind of triangle is \begin{align*}ABC\end{align*} in this case?
7. What kind of a triangle guarantees that the incenter is on the inside of the triangle?
8. Measure the distance from the incenter to each side of the triangle. What relationship is true about the distances?
Problem 3 – Extension
A family purchases a house with the plot given below. The deed states that the backyard of their property is from Fence 2 to Triangle Pond, and equidistant from Fence 1 and Fence 2. The family would like to build a fence around their property. (Assume that the backyard of the property starts at the horizontal axis.)
9. Find at least two possible coordinates for fence posts for the new fence. Keep in mind that the new fence is equidistant from Fence 1 and Fence 2. Round your answer to the nearest tenth.
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