Right Angles and Ratios Slideshow 40, Mathematics Mr. Richard Sasaki

Objectives Understand common types of right-angled triangles See how ratios relate for given angle sizes Be able to show that a triangle is right-angled Be able to show whether other triangles are acute or obtuse

Right-Angled Scalene Triangles A right-angled triangle could be or . isosceles scalene A scalene triangle is a triangle where no lengths or angles are equal. A commonly known scalene triangle is the 30-60-90 triangle. Here, the angles are in ratio 1:2:3 and the edges are in the ratio 1: 3 :2. 30 𝑜 The purple triangle is also 30-60-90 and edges are in the same ratio. 2 5 3 3 5 60 𝑜 60 𝑜 30 𝑜 1 10

Right-Angled Isosceles Triangles The only type of isosceles right-angled triangle is the triangle. 45-45-90 An isosceles triangle is a triangle where a pair of lengths and angles are equal. Here, the angles are in ratio 1:1:2 and the edges are in ratio 1:1: 2 . 45 𝑜 2 The purple triangle is also 45-45-90 and edges are in the same ratio. 1 4 2 4 45 𝑜 4 1

7 2 𝑐𝑚 6 45 𝑜 2 3 30 𝑜 60 𝑜 3 2 30 𝑜 45 𝑜 45 𝑜 3 5 2 3 2 2 3 60 𝑜 3 15 2 6 3 3 𝑐𝑚 3 2𝑎 𝑎 3 𝑎 𝑐𝑚 3 𝑎 5 2 2 𝑐𝑚 5 2 2 𝑐𝑚 𝐴𝑟𝑒𝑎: 𝑏ℎ 2 = 25 4 𝑐 𝑚 2 2.5 2.5

Testing if Right-Angled We know that for a right-angled triangle, 𝑎 2 + 𝑏 2 = 𝑐 2 where 𝑎 and 𝑏 are legs and 𝑐 is the hypotenuse. We can use the opposite principle. If 𝑎 2 + 𝑏 2 = 𝑐 2 is satisfied, the triangle must be right-angled. Example Consider a triangle with edges 15 𝑐𝑚, 36 𝑐𝑚 and 39 𝑐𝑚. Show that it is right-angled. As 𝑎=15, 𝑏=36 and 𝑐=39, 𝑎 2 + 𝑏 2 = 𝑐 2 must be satisfied. 15 2 + 36 2 = 39 2 . This is simplified to make 225+1296=1521 which equates. ∴ the triangle is right-angled.

Yes No No No Yes Yes Two of the numbers are even, so it is not right-angled. As 𝑎=39, 𝑏=80 and 𝑐=89, 𝑎 2 + 𝑏 2 = 𝑐 2 must be satisfied. 39 2 + 80 2 = 89 2 . This is simplified to make 1521+6400=7921 which equates. ∴ the triangle is right-angled. As 𝑎=51, 𝑏=81 and 𝑐=120, 𝑎 2 + 𝑏 2 = 𝑐 2 must be satisfied. 51 2 + 81 2 = 120 2 . This is simplified to make 2601+6561≠14400 which as shown does not equate. ∴ the triangle is not right-angled.

Right-angled Isosceles Acute and Obtuse Triangles An acute triangle exists when all angles are less than 90 𝑜 . An obtuse triangle exists when . one angle is greater than 90 𝑜 A right-angled triangle exists when one angle is exactly 90 𝑜 . Acute Isosceles Obtuse Isosceles Right-angled Isosceles Acute Scalene Obtuse Scalene Right-angled Scalene

Acute and Obtuse Triangles We know that to be right-angled, 𝑎 2 + 𝑏 2 = 𝑐 2 must be satisfied. But how do we show if a triangle is acute or obtuse? When a triangle is acute… 𝑎 2 + 𝑏 2 𝑐 2 > When a triangle is obtuse… 𝑎 2 + 𝑏 2 𝑐 2 < Note: Here there is often no hypotenuse. Allow 𝑐 to represent the longest length or one of the longest lengths.

Acute and Obtuse Triangles Example A triangle has edges 4 𝑐𝑚, 7 𝑐𝑚 and 9 𝑐𝑚. State what type of triangle it is. 𝑎 2 + 𝑏 2 ___ 𝑐 2 4 2 + 7 2 ___ 9 2 16+49___81 65 ___81 < As 𝑎 2 + 𝑏 2 < 𝑐 2 , the triangle must be . obtuse All edges are different so the triangle is obtuse scalene.

(disputed) Obtuse Isosceles Acute Scalene Right-Angled Scalene Acute Scalene Obtuse Scalene Acute Isosceles Right-Angled Scalene Obtuse Scalene Obtuse Scalene The length of the two shorter edges is less than the longest edge.

Obtuse angles are over 90 𝑜 so a triangle with two would be over 180 𝑜 . 𝑎+𝑏>𝑐 All angles are nearly the same. One angle is almost 180 𝑜 and the other two are very small. One angle is small (opposite the base). Other two are nearly 90 𝑜 . Two lengths would be huge in comparison to the third unless one angle is close to 180 𝑜 .