1. Special Right Triangles
30 – 60 – 90
45 – 45 – 90

2. Geogebra Activity Answers

3. 30 – 60 – 90 Triangle
From the Geogebra Activity we know that the “30 – 60 – 90 Triangle” comes from splitting an equilateral triangle in half. Now lets look at a comparison of the sides using the Pythagorean Theorem.

4. 30 – 60 – 90 Triangle
Because this special rt. triangle comes from an equilateral triangle, the hypotenuse must be 2 times as long as the short leg.
Can we use the Pythagorean Theorem to find the long leg?
𝑎 2 + 𝑏 2 = 𝑐 2
2cm
1cm

5. 30 – 60 – 90 Triangle
Therefore, the side-length ratios of : short leg: long leg: hypotenuse are 1 : 3 :2 or 𝑥 :𝑥 3 :2𝑥 These will always hold true for 30 – 60 – 90 Triangles!! Memorize them!!

6. 45 – 45 – 90 Triangle
From the Geogebra Activity we know that the “45 – 45 – 90 Triangle” comes from splitting a square in half from corner to corner. Now lets look at a comparison of the sides using the Pythagorean Theorem.

7. 45 – 45 – 90 Triangle
Because this special rt. triangle comes from a square, both legs must be the same exact length.
Can we use the Pythagorean Theorem to find the hypotenuse?
𝑎 2 + 𝑏 2 = 𝑐 2
1cm
1cm

8. 45 – 45 – 90 Triangle
Therefore, the side-length ratios of : leg: leg: hypotenuse are 1 :1 : 2 or 𝑥 :𝑥 :𝑥 2 These will always hold true for 45 – 45 – 90 Triangles!! Memorize them!!

9. But how do I keep them separate?
30 – 60 – 90 Triangle: 𝟏 : 𝟑 :𝟐 (it’s the one with the 3 in both… “30” and “ 3 ”) 45 – 45 – 90 Triangle: 𝟏 :𝟏 : 𝟐 (the “45’s” repeat and so do the “1’s”)

10. Practice Find the missing sides: Example 1: 5 ft 30
Type of Sp. Rt. Tri. :
Find the missing sides:
Example 1:
30
5 ft

11. Practice Find the missing sides: Example 2: 8 ft 60
Type of Sp. Rt. Tri. :
Find the missing sides:
Example 2:
60
8 ft

12. Practice Find the missing sides: Example 3: 13 cm 45
Type of Sp. Rt. Tri. :
Find the missing sides:
Example 3:
45
13 cm

13. Practice Find the missing sides: Example 3: 20 cm 45
Type of Sp. Rt. Tri. :
Find the missing sides:
Example 3:
45
20 cm