Triangles (A Multiple-Day Lesson)
Objectives
· TSWBAT identify and create triangles based on angle classifications
· TSWBAT identify and create triangles based on side lengths.
· TSWBAT recall and use the Triangle Sum Theorem.
· TSWBAT identify and create special parts of triangles; specifically altitudes, angle bisectors, medians, and perpendiculars
· TSWBAT identify and create special circles related to triangles; specifically the incircle, circumscribed circle, and nine-point circle
Key points
Triangle Classification
Classifications based on angles
·
All three angles are acute in an acute triangle.
·
One angle is an obtuse angle in an obtuse
triangle. It is not possible to have more than one obtuse angle in a triangle.
·
One angle is a right (90°) angle in a right
triangle. It is not possible to have more than one right angle in a triangle.
Classifications based on side lengths
·
No two
sides are congruent (equal in length) in a scalene triangle.
·
At least
two sides are congruent in an isosceles triangle. Also, the two angles opposite
the congruent sides are congruent in an isosceles triangle.
·
All
three sides are congruent in an equilateral triangle. An equilateral triangle
is a more specific case of an equilateral triangle.
Triangle Sum Theorem
· The sum of the measures of the angles of any triangle is equal to 180°.
Special Parts of Triangles
·
Altitudes
– The altitudes of a triangle are formed by “dropping” a perpendicular from
each vertex to the opposite side. The altitudes of a triangle are concurrent,
intersecting in the orthocenter.
·
Angle
bisectors – The angle bisectors of a triangle are formed by the bisectors of
all three angles. The angle bisectors of a triangle are concurrent, intersecting
in the incenter, the center of the inscribed circle.
·
Medians
– The medians of a triangle are formed by connecting the midpoint of each side
to the opposite vertex. The medians of a triangle are concurrent, intersecting
in the centroid, the center of mass of the triangle.
·
Perpendicular
bisectors – The perpendicular bisectors of a triangle are formed by drawing a
line through the midpoint of each side perpendicular to the side. The
perpendicular bisectors are concurrent, intersecting in the circumcenter, the
center of the circumscribed circle.
·
Nine-point
circle – The nine-point circle passes through nine special points on a
triangle; the three midpoints, the feet of the three altitudes, and the
midpoints of the segments connecting the vertices to the orthocenter. Also of
note: the orthocenter, nine-point center, the centroid, and the circumcenter
are collinear, forming the Euler line. Click here for a
JavaSketchpad version of the nine-point circle.
Stations (20 minutes each)
1. Classify the three triangles in GSP as acute, obtuse, or right. Try to change each triangle so that it could be classified differently. What do you notice? How were these triangles created? Investigate by showing the hidden objects. Recreate one of the types of triangles such that it will always be either acute, obtuse, or right. Make sure to thoroughly test your triangle. How can you prove your triangle works?
2. Classify the three triangles in GSP as scalene, isosceles, or equilateral. Try to change each triangle so that it could be classified differently. What do you notice? How were these triangles created? Investigate by showing the hidden objects. Recreate all of the types of triangles such that each will always be either scalene, isosceles, or equilateral. This is surprisingly simpler than the angle classifications. Make sure to thoroughly test your triangles. How can you prove each triangle works? Once you have working models, investigate the angles in the isosceles and equilateral triangles – can you find anything special about their measures and their relationships to each other and the sides?
3. Again, use GSP to investigate the triangles. Measure the angles of all the triangles you can find. What is the sum of the measures of the angles of each triangle? Can you find a triangle with a different answer? Further investigation: what is the sum of the measures of the angles of a quadrilateral? A pentagon? Can you divide larger polygons into triangles to see why this works?
4. To be completed after the first three stations. Journal entry: How can triangles be classified?
Wrap-up (5-10 minutes)
Homework
Return to Carol Love's Lesson Plan