Task 1: Discovering 30 -60 -90 and 45 -45 -90 Triangles

Task 1: Discovering 30°-60°-90° and 45°-45°-90° Triangles a) 30°-60°-90° Triangles The Yield sign pictured to the right is an equilateral triangle with a side length of 2 feet. Adam, a construction manager in a nearby town, needs to check the uniformity of Yield signs around the state and asks for the height (altitude) of your town’s Yield sign.

If you were to split the Yield sign in half vertically, using the altitude as a boundary, you would find yourself with two 30-60-90 right triangles. Let’s look at only the right triangle on the right side. We know that the hypotenuse is 2 ft., what is the length of the short leg?

Use the Pythagorean Theorem to find the length of the third side, or altitude.

The construction manager also needs to know the altitude of the smaller triangle within the sign. Each side of that equilateral triangle is 1 ft. long. Using the same procedure as before, determine the altitude of the smaller triangle.

Now that we have found the altitudes of both equilateral triangles, what patterns do we notice about the data? Fill in the first two rows of the chart below and write down any observations you make. Then, fill in the third row.

Equilateral Triangle Hypotenuse Length Short Leg Length Long Leg Length Side Length

What is the relationship between the lengths of the sides for all 30-60-90 triangles? b) 45°-45°-90° Triangles The baseball diamond shown to the right is, geometrically speaking, a square turned sideways. Each side measures 90 feet. A player is trying to slide into home base, but the ball is all the way at second base! Assuming that the second baseman and catcher are standing on their respective corners of the field, how far does the second baseman have to throw the ball to get it to the catcher?

If you were to split the diamond in half vertically, you would have two 45-45-90 right triangles. Let us examine the triangle on the right. You know that the two legs are 90 feet each. Using the Pythagorean Theorem, find the length of the hypotenuse, or the displacement of the ball. Leave your answer in radical form.

The catcher tags the runner out before he gets to home base. He throws the ball back to the pitcher. What is the displacement of the ball for this throw?

Now that we have found the side lengths of two 45-45-90 triangles, what patterns can we notice about the data? Fill in the first two rows of the chart below. Using your observations from the first two rows, fill in the third row.

Leg Length Other Leg Length Hypotenuse Length

90 ft.

45 2 ft.

20 ft.

What is the relationship between the lengths of the sides for all 45-45-90 triangles?