Discovering 30-60-90 Special Triangles

We begin with an equilateral triangle with 1 unit on each side We begin with an equilateral triangle with 1 unit on each side. This is shown below. Find the measurement of all the angles. Construct an altitude from the top vertex to the base in the above square. The diagonal creates two smaller triangles in your above triangle. Find the angle measures of the angles in the two triangles and the new length of the base of the two triangles. Using the Pythagorean Theorem, find the length of the altitude (or the hypotenuse of the right angle triangle). KEEP IN RADICAL FORM.

We begin with an equilateral triangle with 2 unit on each side We begin with an equilateral triangle with 2 unit on each side. This is shown below. Find the measurement of all the angles and the lengths of each side. Construct an altitude from the top vertex to the base in the above square. The diagonal creates two smaller triangles in your above triangle. Find the angle measures of the angles in the two triangles and the new length of the base of the two triangles. Using the Pythagorean Theorem, find the length of the altitude (or the hypotenuse of the right angle triangle). KEEP IN RADICAL FORM.

We begin with an equilateral triangle with 3 unit on each side We begin with an equilateral triangle with 3 unit on each side. This is shown below. Find the measurement of all the angles and the lengths of each side. Construct an altitude from the top vertex to the base in the above square. The diagonal creates two smaller triangles in your above triangle. Find the angle measures of the angles in the two triangles and the new length of the base of the two triangles. Using the Pythagorean Theorem, find the length of the altitude (or the hypotenuse of the right angle triangle). KEEP IN RADICAL FORM.

Equilateral with ____ Units Side length of Hypotenuse Side length of Base Side Length of Altitude   4 8 10 12 16

Do you see a pattern. Explain Do you see a pattern? Explain. Conclusion: Given an equilateral triangle with “x” unit on each side, what would be the lengths of the sides, and the lengths of the altitude?

8.3: Special Right Triangles

Equilateral Triangle Relationship   Equilateral Triangle Relationship A 30° – 60° – 90° triangle is another special right triangle. You can use an equilateral triangle to find this relationship. When the altitude is drawn from any vertex of an equilateral triangle, two congruent 30° – 60° – 90° triangles are formed. In the figure shown, ∆ 𝐴𝐵𝐷≅∆𝐶𝐵𝐷, so 𝐴𝐷 ≅ 𝐶𝐷. If CD = x, then AC = 2x. This leads to the next theorem. 30˚-60˚-90˚ Triangle Theorem Hypotenuse = 2s Base = 𝑠 3 Altitude = s

Example 4: Find x and y. a) 𝑥 30 5 60 𝑥 3 5 3 90 2𝑥 10 2𝑥=10 𝑥=5

Example 4: Find x and y. b) 𝑥 2𝑥 30 60 𝑥 3 90 4 3 3 ∗ 3 = 4 3 4 3 3 4 3 3 ∗ 3 = 4 3 4 3 3 𝑥 30 60 4 3 𝑥 3 8 3 3 90 2𝑥

Example 4: Find x and y. c) 𝑥 3 =4 3 𝑥=4 𝑥 30 4 60 𝑥 3 4 3 90 2𝑥 8

Example 6: Shaina designed 2 identical bookends according to the diagram below. Use special triangles to find the height of the bookends. 𝑥 30 5 60 𝑥 3 5 3 90 2𝑥 10

Example 7. An equilateral triangle has a side length of 10 inches. Find the length of the triangles altitude. 𝑥 30 5 30 10 60 𝑥 3 5 3 60 90 90 2𝑥 10

Example 8. The altitude of an equilateral triangle is 18 inches. Find the length of a side. 𝑥 3 =18 𝑥=6 3 𝑥 30 6 3 30 60 𝑥 3 18 18 60 90 90 2𝑥 12 3

Summary! Find all the missing side lengths. Leave answers in simplified radical form. 5 6 ∗ 2 5 12 10 3 𝑥 45 8 𝑥 45 5 6 45 𝑥 8 45 𝑥 5 6 90 𝑥 2 8 2 90 𝑥 2 10 3

Summary! Find all the missing side lengths. Leave answers in simplified radical form. 2𝑥=4 3 𝑥=2 3 𝑥 𝑥 3 2 3 ∗ 3 6 𝑥 30 5 30 2 3 60 60 6 𝑥 3 5 3 𝑥 3 90 2𝑥 10 90 2𝑥 4 3