Exploring Centers of Triangles

Assignment 4

By Amber Candela

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Let's Explore Triangles!

Using the explorations and Geometer's Sketchpad, explore properties of triangles.

Explorations were adapted from T³ Geometry Professional Development Confernce

Definitions:

In order to complete the investigations, you should know the following definitions

Altitude of a Triangle

Median of Triangle

Angle Bisector

Perpendicular Bisector

Exploration 1

Which of the three segments (altitude, median or angle bisector) will divide the triangle into two equal areas?

- Draw triangle ABC and from one vertex construct an altitude, a median, and an angle bisector
- Create triangle interiors and compare the areas
- Drag a vertex and make a conjecture
- Does it matter if the triangle is acute? Obtuse?
- Explain wy the areas are equal for one of the segments drawn
- Do equal areas imply congruency? Explain

Exploration 2

Draw triangle ABC and construct an angle bisector of angle ABC.

- Draw the point of intersection (D) of the angle bisector and the opposite side of the triangle from which the angle bisector was drawn.
- Find the length of the two parts formed by the point of intersection, AD and DC.
- Find the length of the two sides that form the bisected angle, AB and BC.
- Calculate the ratios of the parts AD/DC and the ratios of the corresponding sides AB/BC.
- Drag the vertices and make a conjecture about these ratios
- Is the conjecture true for all type of triangles? Acute, right, obtuse? Explain

Exploration 3

Draw and label Triangle ABC and construct the midpoints of the three sides of the triangle

- Construct the three medians of the triangle
- Investigate apparent relationships that exist as a result of the medians being drawn in the triangle
- Manipulate the triangle by dragging different vertices around
- Do any of the relationships that you observed still remain true for all of the different triangles that you observed
**The medians meet at a point called the centroid**- Find at least two properties that appear to be true about the centroid.

Exploration 4

Construct triangle ABC and construct the perpendicular bisectors of each side.

- The perpendicular bisectors meet at a point called the circumcenter
- Find at least two properties that appear to be true about the circumcenter of a triangle
- Test your conjectures by dragging the vertices of the triangle to see if the relationships remain true.
- Measure the distance from the circumcenter to each vertex of the triangle
- Explain and demonstrate why this point is called the circumcenter

Exploration 4

Construct triangle ABC and construct the three altitudes

**Their point of concurrency is called the orthocenter of the triangle**- Find at least two properties that appear to be true about the orthocenter
- Test your conjectures by manipulating the triangle to se if the relationships remain true

Exploration 5

Construct triangle ABC and bisect each of its three angles

**The angle bisectors meet at a point called the incenter**- Find at least two properties that appear to be true about the incenter
- Measure the perpendicular distance from the incenter to each side of the triangle
- Explain and demonstrate why this point is called the incenter

Exploration 6

Construct triangle ABC

- Construct and label the centroid, circumcenter, orthocenter, and incenter of this triangle.
- Hide construction lines after you have found each particular point
- Under what conditions will all of these points coincide? Explain why this happens
- Three of these points will always have something in common, see if you can discover this relationship by studying the points as you drag the vertices of the triangle on the screen
- Investigate the midpoint between the orthocenter and the circumcenter
- Using the midpoint between any vertex and the orthocenter as a radius point, draw a circle centered at the midpoint between the orthocenter and the circumcenter.
- What other points seem to be on circle? Explain.

Triangle Centers Script Tools

- Each of the following triangle centers has a script tool below it. Open the script tool and then create a new document. Use your script tool to create triangles and complete your own investigations

**Circumcenter:** The intersection of the perpendicular bisectors of the sides.

**Orthocenter:** The intersection of the altitudes of the triangle

**Incenter:** The intersection of the angle bisectors

**Centroid:** The intersection of the medians