6.4: Hey Ortho! What’s your Altitude?
This activity is intended to supplement Geometry, Chapter 5, Lesson 4.
Problem 1 – Exploring the Altitude of a Triangle
1. Define Altitude of a Triangle.
Draw the altitudes of the triangles in the Cabri Jr. files ACUTE, OBTUSE, and RIGHT and then sketch the altitudes on the triangles below. To do this, start the Cabri Jr. application by pressing APPS and selecting CabriJr. Open the file ACUTE by pressing \begin{align*}Y=\end{align*}
2. Draw the altitudes for \begin{align*}\triangle{ABC}\end{align*}
3. Fill in the blanks of the following statements about whether the altitude of a triangle is inside, outside, or on a side of the triangle.
a. For the acute \begin{align*}\triangle{ABC}\end{align*}
b. For the obtuse \begin{align*}\triangle{DEF}\end{align*}
c. For the right \begin{align*}\triangle{GHJ}\end{align*}
Problem 2 – Exploring the Orthocenter
Open the file TRIANGLE. You are given \begin{align*}\triangle{ABC}\end{align*}
4. What do you notice about the altitudes of all three vertices?
5. The point of concurrency for the altitudes is the orthocenter. Create and label this point \begin{align*}R\end{align*}
6. Can you move vertex \begin{align*}B\end{align*}
7. Can you move vertex \begin{align*}B\end{align*}
Problem 3 – Exploring the Altitude of an Equilateral Triangle
Open the file EQUILATE. You are given an equilateral triangle \begin{align*}ABC\end{align*}
8. Use the Calculate tool to calculate \begin{align*}EP + FP + GP\end{align*}
Position |
\begin{align*}1^{st}\end{align*} |
\begin{align*}2^{nd}\end{align*} |
\begin{align*}3^{rd}\end{align*} |
\begin{align*}4^{th}\end{align*} |
---|---|---|---|---|
\begin{align*}BD\end{align*} |
||||
\begin{align*}EP+FP+GP\end{align*} |
9. What is the relationship between the measurements of \begin{align*}BD\end{align*}
10. Complete the following statement: The sum of the distances from any point in the interior of an equilateral triangle to the sides of the triangle is ________________.
Problem 4 – Exploring the Orthocenter of a Medial Triangle
The medial triangle is the triangle formed by connecting the midpoints of the sides of a triangle.
Open the file MEDIAL2. You are given a triangle, its medial triangle, and the orthocenter of the medial triangle.
11. What triangle center (centroid, circumcenter, incenter, or orthocenter) for \begin{align*}\triangle{ABC}\end{align*}
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