Lesson 2: Pythagorean Theorem Problem Solving - Extracting a Triangle from a Picture

   Constructing Knowledge

There are times when extracting a triangle from a large picture is required. Break the picture into a combination of smaller, simpler shapes. In this section, you will need to focus on seeing right triangles.

Remember to follow these steps in solving word problems.

  1. Identify the given and the required values.
  2. Draw a diagram if one isn't provided. Label the diagram with the given information.
  3. Write the formula using the information labelled on the diagram.
  4. Solve for the unknown.
  5. Review the answer to ensure it makes sense. (Always verify that the hypotenuse is the longest side).

EXAMPLE

A hopper bin on a farm is used to store grain. It has a cone-shaped bottom and a cone-shaped top. The two cones have different depths.
The diameter of the bin is 16 feet. The slant height of the bottom cone is 12 feet. What is the depth of the bottom cone, to the nearest tenth of a foot?

Solution


Step 1: Identify the given and the required values.

  • The diameter of the bin is 16 feet.
  • The slant height of the bottom cone is 12 feet
  • The depth of the bottom cone is unknown.

Step 2: Draw and label the diagram.

Image Source: Pixabay



Extract the portion of the cone needed to solve for the depth. Label the new diagram. Note: Divide the diameter by 2 to get the radius, which is needed to form the right triangle.



Steps 3 and 4: Write the formula, substitute known values, and solve.

\(\begin{align} x^2+r^2&=s^2 \\ \\ x^2+8^2&=12^2 \\ \\ x^2+64&=144 \\ \\ x^2+\cancel{64-{\color{red}{64}}}&=144-{\color{red}{64}} \\ \\ x^2&=80 \\ \\ \sqrt{x^2}&=\sqrt{80} \\ \\ x&=8.9 \end{align}\)

The bottom cone is approximately 8.9 feet deep.

Step 5: Review the answer

The depth of the bottom cone is less than the cone's slant height (hypotenuse), so the answer is reasonable.


Now, it is your turn! Complete the questions in your Chapter 3, Lesson 2 Practice Makes Perfect that refer to Extracting a Triangle from a Picture.



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