1. Day 99 – Trigonometry of 45-45-60 right triangle 2

2. Introduction
In trigonometry, we frequently encounter angle measures that are multiples of 30°, 45°, and 45°. It is indispensable to study the trigonometric ratios of these special angles based on a right triangle bearing a combination of these angles. We have learned how to find the trigonometric ratios of both the 30−60−90 and 45−45−60 right triangles in our previous lessons. We have also learned how to solve a 30−60−90 right triangle. In this lesson, we will learn how to solve a 45−45−60 right triangle.

3. Vocabulary
right triangle A special type of right triangle that has two congruent acute angles, each measuring 45° and the two legs have equal length. It has angles of 45°, 45° and 90°. 2. Leg Either of the two sides of a right triangle that meet to form a right angle and each of them is opposite an acute angle.

4. Properties of the 45−45−90 Right triangle
1. This is a special right triangle that has two acute angles, each measuring 45° and one angle measuring 90°, the right angle. For this reason, it is also referred to as an isosceles right triangle. 2. It has two equal legs and just like other right triangles, the hypotenuse is the longest side and it is opposite the right angle. This implies that if the length of one leg is given then, the length of the other leg is also known.

5. The 45−45−60 right triangle can be visualized to be in a square of length 𝑥 units as shown below. The hypotenuse is the diagonal of the square and the two legs form the sides of the square as shown below. 𝑥
2 𝑥
45°

6. The hypotenuse is found using the Pythagorean theorem
The hypotenuse is found using the Pythagorean theorem. Hypotenuse 2 = 𝑥 2 + 𝑥 2 =2 𝑥 2 ∴Hypotenuse=𝑥 2

7. Since all 45−45−90 right triangles are similar due to AA similarity criterion, then their sides lengths are proportional. We can solve a 45−45−90 right triangle using the relationship between the sides, even when only one side is known. If the length of one leg is 𝑥 units, the length of the other leg will also be 𝑥 units. The hypotenuse will be equivalent to 𝑥 2 units as shown below. Note: This relation only works for 45−45−90 right triangles.

8. This shows that ratio of the sides is given as: Leg:leg:Hypotenuse=𝑥:𝑥:2 𝑥 This shows that the length of side can be found using this relationship between the sides of the 45−45−90 right triangle. 𝑥
2 𝑥
45°

9. If the only the hypotenuse is given, it possible to find the length of the other two legs. From the 45−45−90 right triangle above, the hypotenuse, 𝐡=𝒙 𝟐 , therefore is we wish to find the length of either side 𝑥, we have: ℎ=𝑥 2 ∴𝒙= 𝐡 𝟐 Multiplying both sides of 𝐡 𝟐 by 2 2 we have:

10. 𝑥= h 2 × 2 2 = h 2 2 = 1 2 h 2 We can use either of the two equations below to find the length of a leg when given the length of hypotenuse only. 𝒙= 𝐡 𝟐 or 𝒙= 𝟏 𝟐 𝐡 𝟐 where 𝑥 is the length of either leg and ℎ is the length of the hypotenuse.

11. In a nutshell, to solve a 45−45−90 right triangle;
The two legs are equal in length.
Length of hypotenuse= 2 Length of either leg
Length of either leg= Length of hypotenuse
Length of either leg= Length of hypotenuse 2

12. Example Find the length 𝑥 and 𝑦 in the triangle below. 12 𝑦
45°

13. Solution This is a 45−45−90 right triangle
Solution This is a 45−45−90 right triangle. 𝒙=𝟏𝟐 The two legs in a 45−45−90 right triangle are equal. We use the relation, ℎ=𝑥 2 to find the length of the hypotenuse, ℎ. ℎ=𝑥 2 ⟹ℎ=12 2 ∴𝒉=𝟏𝟐 𝟐

14. homework
Find the length 𝑥 and 𝑦 in the triangle below. 𝑥 15 𝑦
45°

15. Answers to homework
𝑥= and 𝑦=

16. THE END