Geometry and Scale: Reasoning and Proof at Work

Problem to think about An isosceles triangle of area 100 in 2 is cut by a line parallel to its base into an isosceles trapezoid and a smaller isosceles triangle. The trapezoid has area 75 in 2. Question If the altitude of the triangle is 20 in, what is the length of the “cut line”? MathCounts, 2004 A B E C D ?

Problem to think about Possible approaches? –Compute lengths of all sides of ABE, use similar triangles, … –Use formulas for areas of triangles and trapezoid –Use proportions and scaling formulas for lengths and areas Let’s start with simpler questions MathCounts, 2004 A B E C D ?

Scaling Distances Think about maps at different scales At 1 in : 1 mile, a 10 mile straight road measures 10in. How long is that road on a 1 in : 5 miles scale map? How long is that road on a 1 in : 2 miles scale map?

Scaling Distances & Areas What would be the linear scale factor if the area of the map is to be halved? –4 units wide becomes approx 2.8 units By what factor would you increase the sides of a square in order to double its area? What happens to the area if we double the side lengths?

Scaling Lengths & Areas By what factor would you increase the radius of a circle in order to double its area? What about doubling the circumference? Let’s make a table What happens with circles, squares? Rectangles – TV screen, HDTV? Parallelogram? Trapezoid? Triangle? Spreadsheet: Areas.xlsAreas.xls

Parallelogram

Area = Base x Height What happens if we scale the whole parallelogram keeping proportions?

Areas scale like Square of length scale factor What does this mean for our original problem?

Problem to think about An isosceles triangle of area 100 in 2 is cut by a line parallel to its base into an isosceles trapezoid and a smaller isosceles triangle. The trapezoid has area 75 in 2. Question If the altitude of the triangle is 20 in, what is the length of the “cut line”? MathCounts, 2004 A B E C D ?

Data MathCounts, 2004 A B E C D ?

A B E C D ?

Summary Review of basic area formulas Effect of scaling on length and area Squaring numbers Power of mathematical reasoning A little thought can save a lot of detailed difficulty Note: we are still “problem-solving” Content Strands Geometry, and Algebra Number sense & operations Measurement Process Strands Reasoning & proof, and Problem solving Connections (maps) Others?