6.4: Hey Ortho! What’s your Altitude?
This activity is intended to supplement Geometry, Chapter 5, Lesson 4.
ID: 11483
Time Required: 45 minutes
Activity Overview
In this activity, students will explore the altitudes of a triangle. Students will discover that the altitude can be inside, outside, or a side of the triangle. Students will discover that the altitudes are concurrent. The point of concurrency is the orthocenter. Students should discover the relationship between the type of triangle and the location of the point of concurrency. Students will discover properties of the orthocenter in equilateral triangles.
Topic: Triangles & Their Centers
- Altitudes of a Triangle
- Orthocenter
Teacher Preparation and Notes
- This activity was written to be explored with the Cabri Jr. on the TI-84.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
- To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=11483 and select ACUTE, EQUILATE, MEDIAL2, OBTUSE and TRIANGLE.
Associated Materials
- Student Worksheet: Hey Ortho! Whats your Altitude? http://www.ck12.org/flexr/chapter/9690, scroll down to the fourth activity.
- Cabri Jr. Application
- ACUTE.8xv, OBTUSE.8xv, TRIANGLE.8xv, EQUILATE.8xv, and MEDIAL2.8xv
Problem 1 – Exploring the Altitude of a Triangle
Students should define the altitude of a triangle using their textbooks or other source on their worksheet.
In files ACUTE and OBTUSE, students are to create the altitude of an acute triangle, obtuse triangle, and right triangle, respectively. Students will need to construct the perpendicular line to a line through a point (press ZOOM and select Perp.).
To show the altitude of the triangle, students will need to find the intersection point of the perpendicular line and the line that extends from the base of the triangle. Students will then need to create a segment from the opposite vertex to the intersection point.
Problem 2 – Exploring the Orthocenter
In file TRIANGLE, students are given \begin{align*}\triangle{ABC}\end{align*}. They should construct the altitude of each vertex of the triangle. Students should realize that they are concurrent. Explain to students that the point of concurrency is called the orthocenter of the triangle.
Students should discover a few facts about the orthocenter. If a triangle is an acute triangle, then the orthocenter is inside of the triangle. If a triangle is a right triangle, then the orthocenter is on a side of the triangle. If a triangle is an obtuse triangle, then the orthocenter is outside of the triangle.
Problem 3 – Exploring the Altitude of an Equilateral Triangle
Students will need to find the distance from point \begin{align*}P\end{align*} to the 3 sides of the triangle and the altitude \begin{align*}BD\end{align*} in the file EQUILATE. The Distance & Length tool is found in the GRAPH/\begin{align*}F5\end{align*} menu (press GRAPH and select Measure > D.&Length). Once students make their calculation, they can move it to the right of the appropriate label (\begin{align*}EP\end{align*}, for instance). Students will then need to calculate the sum of \begin{align*}EP + FP + GP\end{align*} using the Calculate tool (press GRAPH and select Calculate). Students will need to move their cursor to \begin{align*}EP\end{align*} until it is underlined, then press ENTER, then press +, then move their cursor to \begin{align*}FP\end{align*} until it is underlined, then press ENTER. A measurement that you can place anywhere on the screen is given. Place the measurement by pressing ENTER. Repeat this process for \begin{align*}EP + FP\end{align*} that was just created and \begin{align*}GP\end{align*} to get the sum \begin{align*}EP + FP + GP\end{align*}. Students should discover that the sum of the distances from all three sides to point \begin{align*}P\end{align*} and the length of the altitude \begin{align*}BD\end{align*} are equal. Make sure that students see that this only works for an equilateral triangle.
Problem 4 – Exploring the Orthocenter of a Medial Triangle
In the file MEDIAL2, students are given a triangle, its medial triangle, and the orthocenter of the medial triangle. Students are to discover which point of the original \begin{align*}\triangle{ABC}\end{align*} is the orthocenter of the medial triangle. Choices should include the centroid, circumcenter, incenter, and orthocenter.
Students should discover that the orthocenter of the medial triangle is the circumcenter of the original triangle.
Solutions
1. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the side opposite that vertex.
2.
3. a. inside
b. outside
c. a side of
4. They are concurrent
5. Right Triangle
6. Acute Triangle
7. Obtuse Triangle
8.
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*} | 4.9 | 5.4 | 5.4 | 5.4 |
\begin{align*}EP+FP+GP\end{align*} | 4.9 | 5.4 | 5.4 | 5.4 |
9. They are equal
10. equal to the length of the altitude
11. circumcenter
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