1. A baseball diamond is square. Each side of the square is 90 feet long. How far is it from home plate to second base?

(a)Draw a diagram of this scenario, complete with labels:

(b)Use the Pythagorean Theorem to find the distance from home plate to second base, both in exact form and rounded to the nearest tenth:

Since this is a square, at first base, we have a right angle and the distance from home plate to second base is the hypotenuse of a right triangle. Now each side is 90 feet and they are both the a and the b in the Pythagorean Theorem.

2. Use the Pythagorean Theorem to find the unknown side of a triangular park which has the following shape. Round to the nearest foot.

Examples: The triangles below are similar. Find the length of the unknown angles and sides of each. Round angles to the nearest degree and sides to the nearest tenth.

Solution:

Since the triangles are similar, we know that all corresponding angles are equal. This means [latex]\angle A = 50^o[/latex] and [latex]\angle W = 84^o[/latex]. Next, since the angles in a triangle add up to [latex]180^o[/latex], we can compute the measure of the third angle in each triangle. In particular, we need to have [latex]50^o + 84^o + \angle V = 180[/latex]. Solving for [latex]\angle V[/latex] gives us [latex]\angle V = 46^o[/latex].

To compute the side lengths, we need to determine the scale factor between the two triangles. Since corresponding sides of 7m and 11.2 m are given, we see that [latex]\frac{7}{11.2}[/latex] is the relationship between each corresponding side lengths. Thus we have:

[latex]\frac{7}{11.2} = \frac{8.2}{x} = \frac{c}{16.2}.[/latex]

Solving for [latex]x[/latex] and [latex]c[/latex] gives us,

[latex]x \approx 13.2[/latex] and

[latex]c \approx 10.125[/latex].

2.

Similar triangles provide a way to find the height of tall objects, such as statues, flagpoles, or trees. At any moment during the day, the sun’s rays cast the same angle between the ground and an object (such as a tree). This means that the “triangles” formed from a tree (or flagpole, etc), the shadow that is cast, and the point from the top of the shadow to the top of the object form similar triangles.

A 120-foot tall tree casts a 104.4-foot shadow. How tall is a tree that casts a 72-foot shadow?

Solution: Since the triangles are similar (as explained above) we can use the proportions between corresponding side lengths to find the missing values.

3.

A 6-foot tall man casts a shadow of 10 feet. How tall is a tree which has a 90 ft shadow measured at the same time?

Solution: Similar to the Example 2 above, we set up and solve an equation using the proportions in the triangles.