Understanding Ratios of Areas of Inscribed Figures Using Interactive Diagrams

This example illustrates how students, using dynamic and interactive geometric figures, can understand connections between algebra and geometry, as described in the Connections Standard. They can develop an understanding of how to justify geometric relationships in a technological environment, as described in the Geometry Standard.

Task

1. Explore the relationship between the area of a triangle and the area of a second triangle whose vertices are the midpoints of the sides of the first triangle. Drag the vertices of the larger triangle to change its shape and size. What do you notice about the ratio of the areas of the two figures? Can you explain why this relationship holds?

[How to Use the Interactive Figure]

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[Stand-alone applet]

2. What kind of relationship is there between the areas of quadrilaterals, similarly formed? Choose the four-sided polygon. Move the vertices around to explore whether the relationship you found in task 1 will still hold. How might you need to adjust your conjecture? Prove your result.
   
   
3. What kind of relationship is there between the areas of n-gons (for n = 5), similarly formed? Predict what will happen if you change the number of sides to five. Then explore whether your conjecture holds. (Be sure to move more than one vertex around.) Will it hold for figures with more than five sides? Amend your conjecture, if necessary, and explain.
   
   
4. How do your results generalize when the vertices of the second figure are points of trisection instead of midpoints? Points of n-section? Return to the triangle and set the slider setting to 0.333. How does this affect the inscribed triangle? What happens to the ratio of the areas? Will it be constant? Explore what happens with other numbers of sides and slider settings.

Discussion

This example gives students an opportunity to explore an interesting quantitative relationship between the areas of a polygon and an inscribed figure formed by joining the midpoints of the polygon's consecutive sides. Students are then given opportunities to explore ways in which those relationships change or remain constant as the number of sides, or the ratio in which the vertices of the inscribed figure cut the sides of the original figure, changes.

In the first task, students will notice that the ratio of the two areas is constant at 0.25. Using their knowledge of similar triangles, they could prove why this relationship holds; see the discussion in the 9–12 Geometry Standards (pages 311–12).

In task 2, when students extend their exploration to quadrilaterals, they may note that a common ratio exists, but instead of being 0.25, it is 0.5. They could again use similar triangles to justify this relationship. Drawing diagonals in the inscribed quadrilateral, they can note that the sum of the areas of the two opposite triangles will be 1/4 of the area of the entire quadrilateral, which is also true for the other pair of opposite triangles. Thus, the areas of the four outer triangles together are one-half the area of the quadrilateral.

In task 3, students may expect that the relationship will extend to polygons with five sides. Some may predict that the ratio for pentagons will be 3/4, or perhaps 5/8. However, when they explore several pentagons using the applet, they will soon find that the ratio is not constant. Nor does a constant ratio emerge for polygons with more than five sides. Investigating how the method using similar triangles would extend to pentagons may help them see why a common ratio cannot be found in that case.

In task 4, students may note that when the slider setting is changed to 0.333, the vertices of the inscribed triangle split the sides of the original triangle in almost a 1:3 ratio, thereby approximating the trisection points. As the vertices of the outer triangle are moved, the ratio of the areas again remains constant, this time at 0.333. When the triangle is equilateral, notice that three 30-60-90 triangles are formed, and some calculations can reveal the desired relationship; it is less clear why the relationships hold in polygons with more than three sides. Changing the number of sides corroborates the results of the previous task: the ratio of areas is preserved for polygons with four or fewer sides but not for polygons with more than four sides.

As the value of the slider changes, the ratio of the areas remains constant for triangles, although the exact relationship between the slider value and the ratio is not obvious. Students may wish to graph the relationship between the slider value and the corresponding ratio of the areas for various slider values, thus revealing a parabola. Substituting values into the general quadratic equation and solving the resulting system of equations to find values for the constants produces an equation that exactly predicts the values. This may seem quite astounding, especially since the reason for this relationship is not obvious. An exploration of the relationship between slider values and the ratio of the areas for quadrilaterals reveals a similar quadratic relationship.

This problem provides a rich context for exploring some surprising geometric relationships. The generalizations about triangles and quadrilaterals can be established by proofs that are within the reach of high school students; their surprising nature may lead students to be curious about why they work. The context also allows for interesting explorations of a situation that can be extended in a number of different ways. Observations made in particular cases may not extend to additional cases, helping students see the danger of making premature generalizations. Finally, students who investigate the relationship between the slider values and the ratio of the areas in triangles or quadrilaterals have the opportunity to discover relationships that clearly illustrate the connections between algebra and geometry.

The activities described in tasks 1 and 2 are based on work conducted in the context of the National Science Foundation project TPE 91-55313. The activity described in task 3 was based on the following article: Zbiek, Rose Mary. "The Pentagon Problem: Geometric Reasoning with Technology." Mathematics Teacher 89 (February 1996): 86–90.

Special thanks to Nick Jackiw for his timely work and keen insights in creating this applet and to Key Curriculum Press for allowing the use of JavaSketchpad™.