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Week 11: Geometry - Triangles

Triangle Inside a Rectangle (Printable PDF)

A triangle has two shared vertices and one shared side with a rectangle. The third vertex is anywhere on the side opposite of the shared side (see figures above).

How does the area of the triangle compare with the area of the rectangle? Why do you think this relationship holds?



As seen in Problem Exploration

M6G1. Students will further develop their understanding of plane figures.

Students can use GSP to construct the triangle in a rectangle, find the areas of both and compare.  Then students can manipulate the triangle to see if conjecture holds.

M6A1. Students will understand the concept of ratio and use it to represent quantitative relationships. The area of the triangle to the area of the rectangle is in a ratio of 1:2.  The students set up the area formulas as ratios: Area of the Triangle/Area of Rectangle = (1/2bh)/bh = 1/2.
M7G1. Students will construct plane figures that meet given conditions.

a. Perform basic constructions using both compass and straight edge, and appropriate technology. Constructions should include copying a segment; copying and angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 
Students can use GSP to construct the triangle in a rectangle, and can manipulate the third vertex to devise and test their conjecture. In order to construct the rectangle, students must construct lines that are parallel and perpendicular to given lines in order to ensure that it is indeed a rectangle.
M8G1. Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.

a. Investigate characteristics of parallel and perpendicular lines both
algebraically and geometrically.

b. Apply properties of angle pairs formed by parallel lines cut by a transversal.

d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent.
Students can show congruent triangles by dropping a perpendicular line from the third vertex of the triangle (not shared with the rectangle’s vertices) to the base. To show congruent triangles, extend the sides of the rectangles as lines. These lines are parallel so properties of the angle pairs formed by the parallel lines and the transversal can be used.
M8A1. Students will use algebra to represent, analyze, and solve problems.

b. Simplify and evaluate algebraic expressions.
Students can use the formulas for area of a triangle and rectangle, and because the height and base are the same value in each formula, the student can simplify to determine a relationship.