Relationships in Triangles
This lesson explores special segments and points of concurrency in triangles using interactive applets. You will investigate perpendicular bisectors, angle bisectors, medians, altitudes, and the points they create: the circumcenter, incenter, centroid, and orthocenter.
Table of Contents
- Section 1: Perpendicular Bisectors & Angle Bisectors
- Section 1a: Perpendicular Bisector
- Section 1b: Angle Bisector
- Section 2: Circumcenter
- Section 2: Circumcenter
- Section 3: Incenter
- Section 3: Incenter
- Section 4: Medians & Centroid
- Section 4a: Triangle Median
- Section 4b: Centroid
- Section 5: Altitudes & Orthocenter
- Section 5a: Triangle Altitude
- Section 5b: Orthocenter
Section 1a: Perpendicular Bisector
Directions for Using the Applet:
1. [b]Look for the gray segment[/b] in the applet.[br][list][*]You’ll see two [b]BIG WHITE POINTS[/b] at each end of the segment.[/*][*][b]Click and drag [/b]these white points to move them around and change the segment.[/*][/list][br]2. [b]Find the slider bar [/b](usually located at the bottom of the applet).[br][list][*]Slide it left and right to observe how the construction changes.[/*][*]You can move the slider by clicking and dragging the dot.[br][br][/*][/list]3. [b]Spend a few minutes interacting[/b] with the applet.[br][list][*]Try different positions for the segment and the slider.[/*][*]Watch carefully what happens to the line that appears when you move things.[/*][/list][br]4. If you’re unsure what to do or want to check your understanding, scroll to the bottom of the page and [b]watch the “Quick (Silent) Demo.”[/b][br]
Question to Answer (Write in Complete Sentences):
[b]What do you think it means for a line to be a perpendicular bisector of a segment?[/b]Use what you observed in the applet to explain your thinking. Be sure to answer in [b]complete sentences[/b].
Quick (Silent) Demo
Section 2: Circumcenter
Directions for Using the Applet:
1. [b]Explore the triangle in the applet[/b] by clicking and dragging the [b]WHITE VERTICES[/b] of the triangle to new positions.[br][br]2. Use the [b]slider[/b] to adjust the construction. Make sure to:[br][list][*]Change the triangle’s shape [b]before[/b] and [b]after[/b] sliding.[/*][*]Try making the triangle acute, right, and obtuse to see how things change.[br][/*][/list][br]3. Pay attention to:[br][list][*]The [b]3 small blue points[/b] that appear on the triangle’s sides.[/*][*]The [b]3 brown lines[/b] connected to those points.[/*][*]Where the brown lines intersect (orange point = [b]Circumcenter[/b]).[/*][*]The [b]purple circle[/b] that appears around the triangle.[/*][*]The [b]pink slider[/b], which controls the angle at the pink vertex in the lower left.[br][/*][/list][br]4.[b] Interact with the applet for a few minutes.[/b][br]Try different triangle shapes and adjust the sliders. Observe how the circumcenter and the circle behave.[br][br]5. If you’re unsure what to do, scroll to the bottom and watch the [b]“Quick (Silent) Demo.”[/b]
Questions to Answer (Write in Complete Sentences):
1. What are the [b]three small blue points[/b] on the triangle? How do you know?
2. What kind of lines are the [b]three brown lines[/b]? What makes you say that?
3. What do you notice about the [b]intersection of the three brown lines[/b]?
4. The [b]orange point[/b] is the [b]circumcenter[/b]. Based on what you observed:[br][br]a) Can the circumcenter ever lie [b]outside[/b] the triangle? If yes, what type of triangle causes this?[br][br]b) Can the circumcenter lie [b]on[/b] the triangle? If yes, what type of triangle causes this, and where is the circumcenter located?[br][br]c) Can the circumcenter lie [b]inside[/b] the triangle? If yes, what type of triangle causes this?
5. What do you notice about the [b]purple circle[/b] and the triangle’s vertices?
6. What [b]previously learned theorem[/b] explains why the distance from the circumcenter to each vertex is always the same?
Section 3: Incenter
Directions for Using the Applet:
1. You can [b]move any vertex[/b] of the triangle by clicking and dragging the white points.[br][br]2. Use the [b]smaller slider[/b] to adjust the angle in the [b]lower left corner[/b] of the triangle.[br][br]3. [b]Play around with both the vertices and the slider:[/b][br][list][*]Try different triangle shapes (acute, right, obtuse).[/*][*]Zoom in or out if needed for a better view.[/*][/list][br]4. Focus on the [b]three black segments[/b] that connect each side of the triangle to the point inside it (the [b]incenter[/b]).[br][br]5. Spend a few minutes interacting with the applet and noticing how the incenter and distances behave as the triangle changes.[br][br]6. If you get stuck or are unsure what to do, scroll down and watch the[b] “Quick (Silent) Demo.”[/b]
Questions to Answer (Write in Complete Sentences):
1. What kind of segments are drawn from the triangle’s [b]vertices to the opposite sides[/b]? What makes you say that?
2. How do these [b]three segments intersect?[/b] Describe what you see.
3. The point inside the triangle is the [b]incenter[/b]. Describe the [b]three equal distances[/b] connected to it.[br][br][list][*]Each distance is from the _________ to the _________?[/*][/list]
4. Why are those distances equal? What [b]previously learned theorem[/b] supports this?
Quick (Silent) Demo
Section 4a: Triangle Median
Directions for Using the Applet:
1. Look for the [b]colored triangle[/b] and a [b]green segment[/b] that appears. This segment is a [b]median[/b].[br][br]2. [b]Click and drag the triangle’s vertices[/b], especially the [b]colored vertex[/b], to explore different triangle shapes.[br][br]3. Watch how the green segment behaves as the triangle changes.[br][br]4. Spend a few minutes exploring and try to figure out what makes the green segment special.[br][br]5. If you need help, scroll to the bottom of the page and watch the [b]“Quick (Silent) Demo.”[/b]
Questions to Answer (Write in Complete Sentences):
1. Can a triangle’s [b]median[/b] ever lie [b]outside[/b] the triangle? Explain why or why not.
2. Based on what you observed, [b]complete this sentence[/b] in your own words:[br][br]A median of a triangle is…[br][br][color=#ff0000]***DO NOT google the term and give me a definition please :)[/color]
Section 5a: Triangle Altitude
Directions for Using the Applet:
1. In this applet, you’ll see a [b]purple segment[/b] appear. This segment is called an [b]altitude[/b] of the triangle.[br][br]2. [b]Click and drag the triangle’s vertices,[/b] especially the [b]blue vertex[/b], to create different triangle shapes.[br][br]3. [b]Slide the slider[/b] to activate the altitude and observe how it behaves.[br][br]4. Watch where the altitude falls in relation to the triangle—it might be [b]inside[/b], [b]on[/b], or [b]outside[/b] the triangle depending on its shape.[br][br]5. Take a few minutes to explore and test different triangle types (acute, right, obtuse).[br][br]6. If you’re unsure, scroll to the bottom of the page and watch the[b] “Quick (Silent) Demo.”[/b]
Questions to Answer (Write in Complete Sentences):
1. Is it ever possible for a triangle’s [b]altitude[/b] to lie [b]inside[/b] the triangle? Explain.
2. Can an altitude ever lie [b]on the triangle itself [/b](for example, as one of the sides)?[br][br]If so, what kind of triangle would make this happen?
3. Is it ever possible for an altitude to lie [b]entirely outside[/b] the triangle? If yes, under what condition?
4. Based on your observations, [b]complete this sentence[/b] in your own words:[br][br]An altitude of a triangle is…[br][br][color=#ff0000]DO NOT google the term and give me a definition :)[/color]