Illuminations: Inequalities in Triangles

# Inequalities in Triangles

 Students will use pasta to create models of triangles and non-triangles and investigate the relationship between the longest side of the triangle and the sum of the other two sides of the triangle. In addition, students will measure the sides and angles of a scalene triangle and investigate the relationship between the location of the largest angle and largest side in a triangle.

### Learning Objectives

 Students will: Investigate the relationship between the largest side and the sum of the remaining sides in a triangle. Investigate the relationship between the largest side and the largest angle in the triangle. Use the triangle inequality to solve problems involving triangles. Use the inequality for sides and angles in a triangle to solve problems involving triangles.

### Materials

 Long, thin pasta (such as spaghetti or linguine) Rulers Protractors The Triangle Inequality Activity Sheet Inequalities for Sides and Angles of a Triangle Activity Sheet

### Questions for Students

 If the sum of the measures of the small and medium sides of the triangle is greater than the measure of the large side of the triangle, why can it be concluded that the sum of the measures of any other pair of sides of the triangle will be greater than the measure of the remaining side? [The sum of any other pair of sides must include the large side and as such must be larger than either the medium side or the small side by itself.] Is it possible to have a triangle having the sum of the measures of the small and medium sides equal to the measure of the large side? [If the sum of the small and medium sides is equal to the long side, the triangle will collapse into a single line segment.] The inequality for sides and angles of a triangle states that the longest side of the triangle must always be opposite the greatest angle of the triangle and that the shortest side of the triangle must always be opposite the smallest angle of the triangle. Why would it be impossible to draw a triangle where the longest side of the triangle was not opposite the greatest angle of the triangle? [Consider a scalene triangle and—without loss of generality—let AB > AC. Extend side AC and locate point D so that AD = AB. Triangle ABD is an isosceles triangle, so m∠ABD = m∠ADB. Also, because ∠ACB is an external angle of triangle BCD, m∠ACB = m∠CDB + m∠DBC. Consequently, m∠ACB > m∠CDB = m∠DBA > m∠CBA. Therefore, by the transitive property, m∠ACB > m∠CBA.]

### Assessment Options

 Two sides of a triangle are 6 cm and 10 cm long. Determine a range of possible measures for the third side of the triangle. [4 cm < x < 16 cm] Given that m∠CDB = 65°, m∠CBD = 72°, m∠ADB = 34°, and m∠A = 87°, list the segments in the diagram in order from longest to shortest. [CD, CB, BD, AD, AB.]

### Extensions

 Discuss the relationship between the triangle inequality and vector addition. Use the following diagram to illustrate: Using the triangle inequality, |x| + |y| ≥ |x+y| . [Notice that if |x| + |y| = |x+y|, the vectors would be pointing in the same direction and the diagram would not form a triangle.]

### Teacher Reflection

 How did the students demonstrate understanding of the materials presented? What were some of the ways that the students illustrated that they were actively engaged in the learning process? Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?

### NCTM Standards and Expectations

 Geometry 9-12Analyze properties and determine attributes of two- and three-dimensional objects. Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.
 This lesson prepared by Heather Godine.

1 period

### NCTM Resources

 More and Better Mathematics for All Students
 © 2000 National Council of Teachers of Mathematics Use of this Web site constitutes acceptance of the Terms of Use The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The views expressed or implied, unless otherwise noted, should not be interpreted as official positions of the Council.