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This section uses Pythagorean triples and similar triangles to solve problems.

Example

Firefighters use a ladder that is 25 meters long to reach windows of buildings above the ground floor. In this case, firefighters need to reach a window that is on the upper story of a house. They lean the ladder against the house, as shown in the diagram below. How high from the ground is the base of the window?

Image is a sketch of the word problem. The right triangle in the sketch as shows that one leg of the triangle is 15 meters, the other leg is unknown so it is represented by X. The hypotenuse of the triangle is 25 meters.

Step 1: Identify the right triangle in the figure by clicking on a part of the figure that represents a side of the right triangle.

Fill in the Blanks

Click on the blanks to reveal the answers.

Step 2: Determine the lengths of the triangle and whether or not the triangle has side lengths that are a Pythagorean triple. If so, then identify the Pythagorean triple.

The sides of the right triangle are: Interactive button. Assistance may be required. _____ The length of the shorter leg is the distance between the base of the ladder and the house, or 15 meters and the length of the hypotenuse is the length of the ladder, or 25 meters.
The Pythagorean Triple is: Interactive button. Assistance may be required. _____ 15, 20, 25, since the length of the shorter leg of the right triangle is 15 meters and the length of the hypotenuse is 25 meters.

Step 3: Decide if it is a Pythagorean triple. Is it a dilated version of a smaller, yet similar, right triangle? If so, state the scale factor used to generate this triangle from the smaller one.

Dilated version of Interactive button. Assistance may be required. _____ A smaller 3-4-5 right triangle
The scale factor is Interactive button. Assistance may be required. _____ 5

Step 4: Relate the sides of the Pythagorean triple right triangle to the diagram of the real-life problem.

5 ⇒ Interactive button. Assistance may be required. _____ 25 m
3 ⇒ Interactive button. Assistance may be required. _____ 15 m
4 ⇒ Interactive button. Assistance may be required. _____ x

Image shows a right triangle with legs of 3 and 4 and a hypotenuse of 5 and the original triangle from the problem.

Step 5: Identify the height of the base of the window.Interactive button. Assistance may be required. _____ The height of the window is 20 meters above the ground and the longer leg of the right triangle.