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Week 9: Geometry - Points, Lines, & Planes

All Swimmed Out (Printable PDF)

Suppose Sammy the swimmer at the tip of pier H wants to swim to the tip of pier I. Pier H is 2 km long and pier I is 1 km long. Since the swim is very long from H to I, Sammy thinks he will need to stop off at the beach to take a break (at point J). Sammy can stop at any point on the beach between the two piers (drag point J around once you construct this in a geometry software or view the java applet below). If Sammy takes the break, where should he stop on the beach if he wants to swim the least distance (the blue path) for the entire trip? What is the shortest distance Sammy can swim for the entire trip? 

**In looking at this problem, the picture implies that the piers are parallel to each other and perpendicular to the beach.  You may want to let the class decide if they want to assume these properties.  These have been assumed in the following chart.



As seen in Problem Exploration

M6A1 Students will understand the concept of ratio and use it to represent quantitative relationships. 

The piers are in a ratio of 2:1 and eventually we will have similar triangles in that same ratio.

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

c. Use the concepts of ratio, proportion, and scale factor to demonstrate the relationships between similar plane figures.

d. Interpret and sketch simple scale drawings.

e. Solve problems involving scale drawings.

Students will be constructing scale drawings of the pier and the beach.  They can use GSP to help determine the minimum distance. 

M7G1 Students will construct plane figures that meet given conditions. (Optional)

a. Perform basic constructions using both compass and straight edge, and appropriate technology.

Students can construct their own scale model using a straightedge and a compass, which involves constructing perpendicular lines and/or parallel lines. Also, the midpoint could be constructed.
M7G3 Students will use the properties of similarity and apply these concepts to geometric figures.

a. Understand the meaning of similarity, visually compare geometric figures for similarity, and describe similarities by listing corresponding parts.

c. Understand congruence of geometric figures as a special case of similarity: The figures have the same size and shape.

With the construction of the scale drawing, one will find that there are similar triangles involved in this investigation. Students can apply the properties of similar figures to determine if the triangles are similar and once they decide that they are similar triangles, they can continue to apply the properties to find missing angle or side measures.

M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation.

a. Find square roots of perfect squares.

f. Estimate square roots of positive numbers.


The student will use the Pythagorean Theorem to solve the problem. The student can estimate the square roots of non-perfect squares before using a calculating devise for more exact calculations.

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.

d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent.

a. By assumption, the piers are parallel to each other and are perpendicular to the beach. Students can further investigate this using GSP.

d. The two triangles created by the point on the beach (where Sammy rests is 2/3 the length between the piers) create two similar triangles, not congruent. The teacher can lead a discussion about the differences in similar and congruent figures.

M8G2 Students will understand and use the Pythagorean theorem.

a. Apply properties of right triangles, including the Pythagorean theorem

By using the properties of right triangles, students can calculate the length of the distance from the pier to a point on the beach by using the Pythagorean Theorem.
M8A1 Students will use algebra to represent, analyze, and solve problems. Students can use a variable to represent the distance of the beach, then calculate what the minimum distance traveled would be.

M8A3 Students will understand relations and linear functions.

i. Identify relations and functions as linear or nonlinear


If solving algebraically, set the distance of the beach to be a fixed number and let the unknown variable be the beach distance from one of the piers to the resting point. Solve using the Pythagorean Theorem for the distances from the end of the piers to the resting point on the beach. Add those distances together, plot and graph to determine a minimum.