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*}, selecting Open..., and selecting the file. Construct the altitude of \begin{align*}\triangle{ABC}\end{align*} on your handheld by pressing ZOOM, selecting Perp., clicking on the side of the triangle, and then clicking on the opposite vertex. Repeat for the files OBTUSE and RIGHT.
2. Draw the altitudes for \begin{align*}\triangle{ABC}\end{align*}, \begin{align*}\triangle{DEF}\end{align*}, and \begin{align*}\triangle{GHJ}\end{align*} below.
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*}, the altitude of vertex \begin{align*}B\end{align*} is ________________ the triangle.
b. For the obtuse \begin{align*}\triangle{DEF}\end{align*}, the altitude of vertex \begin{align*}E\end{align*} is _______________ the triangle.
c. For the right \begin{align*}\triangle{GHJ}\end{align*}, the altitude of vertex \begin{align*}H\end{align*} is _________________ the triangle.
Problem 2 – Exploring the Orthocenter
Open the file TRIANGLE. You are given \begin{align*}\triangle{ABC}\end{align*}. Construct the altitude of each vertex of the triangle. Use your constructions to answer the following questions.
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*}. Is it possible to move vertex \begin{align*}B\end{align*} so that the orthocenter 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*}B\end{align*} so that the orthocenter is inside of \begin{align*}\triangle{ABC}\end{align*}? If so, what kind of triangle is \begin{align*}ABC\end{align*} in this case?
7. Can you move vertex \begin{align*}B\end{align*} so that the orthocenter is outside of \begin{align*}\triangle{ABC}\end{align*}? If so, what kind of triangle is \begin{align*}ABC\end{align*} in this case?
Problem 3 – Exploring the Altitude of an Equilateral Triangle
Open the file EQUILATE. You are given an equilateral triangle \begin{align*}ABC\end{align*} with altitude \begin{align*}\overline{BD}\end{align*} and point \begin{align*}P\end{align*} on the inside of the triangle. Find the distance from point \begin{align*}P\end{align*} to the three sides of the triangle using the Length tool found by pressing GRAPH and selecting Measure > D. & Length. Also, find the length of \begin{align*}\overline{BD}\end{align*} and answer the following questions.
8. Use the Calculate tool to calculate \begin{align*}EP + FP + GP\end{align*}. Move point \begin{align*}A\end{align*} to 2 different positions and record the measurements in the table below. Next, move point \begin{align*}P\end{align*} to 2 different positions and record the measurements in the table below.
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*}BD\end{align*} | ||||
\begin{align*}EP+FP+GP\end{align*} |
9. What is the relationship between the measurements of \begin{align*}BD\end{align*} and \begin{align*}EP + FP + GP\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*} is the orthocenter, \begin{align*}O\end{align*}, of the medial \begin{align*}\triangle{DEF}\end{align*}?
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