InterMath
Title: Triangle Inside a Rectangle
Problem Statement
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?
Problem setup
I am trying to determine the relationship between the area of a triangle inscribed in the rectangle to the area of the rectangle to the area of the rectangle. Note that the triangle shares a side with the rectangle and its third vertex lies on the opposite side of the rectangle.
Plans to Solve/Investigate the Problem
Prediction: ½ because, visually, it looks like ½.
First I plan to construct the figure above in GSP. Then I will measure the area of the triangle and measure the area of the rectangle. I will use GSP’s measurement feature and the area formulas of each shape.
Area of a triangle equals--A=1/2 * base * height
Area of a rectangle equals---A=length * width
I will use these areas in order to determine if there is a constant ratio of their areas. Next, I will begin to make a case for why this is always true.
Investigation/Exploration of the Problem
In order to discuss how I went about solving this problem I will do it in a step by step process.
1. Create a rectangle: draw a line segment, construct 2 lines perpendicular to the line segment, put a point on one of the perpendicular lines and construct a perpendicular line to that point.
2. Highlight two points at a time to make sure each side is a segment.
3. Measure parallel sides of the rectangle to make sure they are equal.
4. To construct the triangle inside the rectangle, I placed a point E on segment BC and constructed a segment AE from point A to point E and constructed a segment ED from point D to point E.
As you can see in this illustration, the base of the triangle would be line segment AD. Line segment AD is also the length of one side of the rectangle. Therefore the base of triangle and the length of the rectangle are equal.
The above illustration shows the relationship between the area of the triangle and rectangle. As I predicted, the area of the triangle is half of the area of the rectangle. The side formulas show the relationship and you can see that the area of the triangle is exactly half of the area of the rectangle. I think this relationship can be better seen in the figure below.
This figure illustrates perfectly why the area of the triangle is half the area of the rectangle. As you can see, the triangle takes up half of the area of the rectangle. Not only that but the area formula for the rectangle is length x width. The area of the triangle is ½ (base x height). In the above illustration, you can see that the base of the triangle and the length of the rectangle are the same as are the width of the rectangle and the height of the triangle. The base and height of the triangle then must be multiplied by ½ to get the area which is why the area of the triangle is always ½ of the rectangle.
Extensions of the Problem
What if we extend the sides of the rectangles to lines BC and AD rather than line segments BC and AD. If the vertex of the triangle can be anywhere on line BC, will the same relationship hold?
The triangle will still be ½ of the rectangle and the relationship will still hold. The triangle will still have the height of line segment AB and will still have the same base of line segment AD. The triangle will have the same height as well as the same base.
The triangle has merely made a transformation where the shape has changed. The area of the triangle will still hold true to the shape. Conservation also comes to mind. Although the triangle has been stretched out for lack of better words, the area still remains the same as before.
Author & Contact
Chantel Lewis
GPS Connection: Standard M5M Measurement/M5M1.
A. Estimate the area of fundamental geometric plane figures
C. Derive the formula for the area of a triangle (demonstrate and explain its relationship to the area of a rectangle with the same base and height.)
D. Find the area of triangles using formulae. Students will extend their understanding of area of fundamental geometric plane figures.
** In order to investigate this problem, students must understand geometric plane figures as well as area of triangles and rectangles.
Link(s) to resources, references, lesson plans, and/or other materials