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Grades 9-12

9.05 Modeling with congruent triangles

Lesson
Practice

Introduction

In Algebra 1, we were introduced to the modeling cycle and how different algebraic models can be used to solve real-world problems. In Geometry, our modeling cycle often relies on a diagram or geometric figure and the properties of that figure:

When creating a geometric model, we will need to:

A modeling cycle. Starting with the phrase Identify the problem inside a circle. An arrow pointing to the right where the phrase Create a model is inside a rectangle. Next is an arrow pointing downward where the phrase Apply and analyze is inside a rectangle. Then, an arrow pointing to the right where the phrase Interpret results is inside a rectangle. Next is an arrow pointing upward where the phrase Verify the model is inside a triangle. Then, an arrow pointing to the right where the phrase Report findings is inside a circle. There is an arrow pointing to the left from the phrase Verify the model to Create a model.

  1. Identify the essential features of the problem

  2. Create a model using a diagram, graph, table, equation or expression, or statistical representation

  3. Analyze and use the model to find solutions

  4. Interpret the results in the context of the problem

  5. Verify that the model works as intended and improve the model as needed

  6. Report on our findings and the reasoning behind them

Creating a geometric model

So far in Geometry we have studied the properties of lines, angles, parallel lines, rigid transformations, and triangles. Now, we want to apply these concepts to realistic design problems and utilize the properties of our chosen figures.

Exploration

A quilt blanket is to be created by copying the following design:

A pattern of triangles. Speak to your teacher for more information.

  1. What do you notice about the quilt pattern?

  2. Explain a strategy, that requires the least amount of measurement, that a quilter could use to determine the amount of each color of fabric they would need.

Congruent triangles can be used to minimize the number of calculations or measurements needed in a design problem. How we view an image in terms of shapes will influence how we approach the problem. In this quilt, we can group the triangles by their shape and size and use one calculation to apply to all triangles of that shape and size.

Examples

Example 1

A construction worker is building the fire escapes for an apartment building, similar to the building shown. He has purchased identical ladders to be installed on each floor.

Eight ladders in a building. Speak to your teacher for more information.

Create a model and use the model to explain a strategy the builder can use to guarantee each level of the fire escape is the same height.

Worked Solution
Create a strategy

We first need to determine what geometric figures will help us model this problem. Some approaches may use parallel lines, perpendicular lines, or congruent right triangles.

Apply the idea

Here's a potential model:

Eight ladders in a building. Speak to your teacher for more information.

We know that the ladders are all congruent since they are all the same ladder.

Let's assume that each floor will be parallel to the ground, and that the height will be measured perpendicular to that.

With these assumptions, the floor, ladder, and height form a right triangle and the hypotenuse of each right triangle (formed by the ladder) will be congruent.

If the construction worker wants to guarantee that each floor is the same height, he can install the ladder at the same angle. Then, the triangles would be congruent by AAS as follows:

  • The first angle is the right angle formed by the height and floor.
  • The second angle is the angle at which the worker installs the ladder.
  • The side is the ladder.

If the triangles are congruent, then the height of each floor is congruent by CPCTC.

Reflect and check

If the worker doesn't have a reliable way to measure the angle he installs the ladder at, the construction worker could use the same length for the landing platform at each level of the fire escape. This would ensure that the triangles formed are congruent by the HL postulate as follows:

  • The hypotenuse is congruent because the ladders are all the same.
  • The leg is congruent because the platforms are all the same length.
  • The triangle is a right triangle because the height is assumed to be perpendicular to the floor.
Idea summary

To create a geometric model, we can begin by identifying familiar figures and assumptions in the real-world problem and use the properties of those figures when we apply our model.

Outcomes

G.SRT.B.5

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

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