Lesson 2: Pythagorean Theorem Problem Solving - Indirect Measurement

   Constructing Knowledge

Previously you combined calculations from multiple right angled triangles within a single shape to figure out a missing dimension. In this section you will combine multiple right angle triangles as a means of measuring a distance. In these types of questions, you will create a plan as to how a distance can be calculated. Often it is helpful to use headings written in words to help organize work, not just the math involved.

EXAMPLE

Kirk was mountain climbing when he injured himself in the location indicated in the picture. Search and rescue came to lower him to a nearby road where an ambulance was waiting.

For each of the three distinct climbing sections Kirk kept track of how much climbing rope he used and the horizontal distance he traveled.

Section Rope used Horizontal distance
I 18 m 8 m
II 10.5 m 8 m
III 16 m 10 m


How high above the road was Kirk when he got injured?

Solution


Step 1: Label the diagram



Step 2: Write a Plan

The distance Kirk was lowered would be equal to the heights of the triangles.

distance lowered = height I + height II + height III

Step 3: Calculate height of I

\(\begin{align} a^2+b^2&=c^2 \\ \\ a^2+8^2&=18^2 \\ \\ a^2+64&=324 \\ \\ a^2+\cancel{64-{\color{red}{64}}}&=324-{\color{red}{64}} \\ \\ a^2&=260 \\ \\ \sqrt{a^2}&=\sqrt{260} \\ \\ a&=16.1\,\text{m} \\ \end{align}\)

The height of I is approximately 16.1 m.

Step 4: Calculate height of II

\(\begin{align} a^2+b^2&=c^2 \\ \\ a^2+8^2&=10.5^2 \\ \\ a^2+64&=110.25 \\ \\ a^2+\cancel{64-{\color{red}{64}}}&=110.25-{\color{red}{64}} \\ \\ a^2&=46.25 \\ \\ \sqrt{a^2}&=\sqrt{46.25} \\ \\ a&=6.8\,\text{m} \\ \end{align}\)

The height of section II is approximately 6.8 m.

Step 5: Calculate height of III

\(\begin{align} a^2+b^2&=c^2 \\ \\ a^2+10^2&=16^2 \\ \\ a^2+100&=256 \\ \\ a^2+\cancel{100-{\color{red}{100}}}&=256-{\color{red}{100}} \\ \\ a^2&=156 \\ \\ \sqrt{a^2}&=\sqrt{156} \\ \\ a&=12.5\,\text{m} \\ \end{align}\)

The height of section III is approximately 12.5 m.

Step 6: Calculate the total height.

Total height = height I + height II + height III
= 16.1 m + 6.8 m + 12.5 m
= 35.4 m

Kirk was approximately 35 m above the road when he got injured.


Now, it is your turn! Complete the questions in your Chapter 3, Lesson 2 Practice Makes Perfect that refer to Indirect Measurement.



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