# Relationship of proportion and similarity

### Ratios, Proportions, and Similarity — Math Similarity Proportion activities for Geometry students. Explore the relationship of perimeter and area in similar triangles when the scale factor is changed. Two geometrical objects are called similar if they both have the same shape, or one has the are in proportion, and corresponding angles of similar polygons have the same . The relationship holds for figures that are not rectifiable as well. Learn about proportional relationships by exploring paint mixtures and studying science experiments on the expansion of heated air. Begin to learn about slope.

The Incredible Shrinking Universe Create scale models of our Solar System and calculate the travel time to another star. Ramps, Paints, and Hot-Air Balloons Learn about proportional relationships by exploring paint mixtures and studying science experiments on the expansion of heated air.

## Advanced Common Core Math Explorations: Ratios, Proportions, and Similarity

Begin to learn about slope by helping a company design multi-purpose ramps. Sample Tasks Exploration 3: Analyze the gears on a speed bike and figure out how to use gear ratios to calculate a bike's speed. Help a cycler design a cycling calculator for her new website. Perplexing Percentages Learn the ins and outs of percentages by solving a set of increasingly challenging real-world problems.

Draw diagrams that connect percentage concepts to proportional relationships, and use algebraic reasoning to make predictions and prove them. Sample Tasks Exploration 5: Scaling a Tower Use scale drawings to help Derek and Beth estimate the height of a transmission tower near their home.

In the process, begin investigating size and angle relationships in similar triangles.

### Similarity (geometry) - Wikipedia

If you continue to the final stage of the activity, you will explore the tangent function from trigonometry and develop strategies for solving a system of algebraic equations.

Keep it in Proportion Discover and explain what causes cross product relationships in proportions.

Help Me Wtih Geometry - Ratio & Proportion and Similarity - Tom English

Explore examples of "tricky" relationships that appear to be proportional but are not. All the corresponding sides have lengths in the same ratio: This is equivalent to saying that one triangle or its mirror image is an enlargement of the other. Two sides have lengths in the same ratio, and the angles included between these sides have the same measure. This is known as the SAS similarity criterion. Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other transitivity of similarity of triangles. Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio. The statement that the point F satisfying this condition exists is Wallis's postulate  and is logically equivalent to Euclid's parallel postulate.

In the axiomatic treatment of Euclidean geometry given by G. Birkhoff see Birkhoff's axioms the SAS similarity criterion given above was used to replace both Euclid's Parallel Postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms.