LESSON EIGHTEEN SPECIAL RIGHT TRIANGLES SPECIAL RIGHT TRIANGLES

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LESSON EIGHTEEN: SPECIAL RIGHT TRIANGLES

LESSON EIGHTEEN: SPECIAL RIGHT TRIANGLES

SPECIAL RIGHT TRIANGLES • In the past lessons we have discussed the Pythagorean Theorem

SPECIAL RIGHT TRIANGLES • In the past lessons we have discussed the Pythagorean Theorem and Special Properties of Triangles particularly the side-angle relationships.

SPECIAL RIGHT TRIANGLES • Today, we are combining those concepts into a new one

SPECIAL RIGHT TRIANGLES • Today, we are combining those concepts into a new one called special right triangles. • The first, we will discuss is a 45 -45 -90. • Why do you think we call it that?

SPECIAL RIGHT TRIANGLES • 45 -45 -90 Triangles are ones which have two 45

SPECIAL RIGHT TRIANGLES • 45 -45 -90 Triangles are ones which have two 45 degree angles and one 90 degree angle. • Would this make it scalene, isosceles or equilateral?

SPECIAL RIGHT TRIANGLES • The theorem that involves the 45 -45 -90 triangle is

SPECIAL RIGHT TRIANGLES • The theorem that involves the 45 -45 -90 triangle is as follows. • In a 45 -45 -90 triangle, the legs (l) are congruent and the length of the hypotenuse (h) is √ 2 times as big. l √ 2 l l

SPECIAL RIGHT TRIANGLES • To find the hypotenuse of a 45 -45 -90, all

SPECIAL RIGHT TRIANGLES • To find the hypotenuse of a 45 -45 -90, all we need is one of the legs. • What are x and y? x 6 y

SPECIAL RIGHT TRIANGLES • What are x and y? 9√ 2 x y

SPECIAL RIGHT TRIANGLES • What are x and y? 9√ 2 x y

SPECIAL RIGHT TRIANGLES • The other type of special right triangle we will discuss

SPECIAL RIGHT TRIANGLES • The other type of special right triangle we will discuss is 30 -60 -90. • This type of triangle has a special theorem too.

SPECIAL RIGHT TRIANGLES • In a 30 -60 -90 triangle, the length of the

SPECIAL RIGHT TRIANGLES • In a 30 -60 -90 triangle, the length of the hypotenuse (h), is 2 times the length of the short leg (s) and the length of the longer leg is √ 3 times the length of the shorter leg. 2 s s s √ 3

SPECIAL RIGHT TRIANGLES • So, from any side of a 30 -60 -90 triangle,

SPECIAL RIGHT TRIANGLES • So, from any side of a 30 -60 -90 triangle, you can find the other two. • Find x and y. x 6 y

SPECIAL RIGHT TRIANGLES • Find x and y. 18 x y

SPECIAL RIGHT TRIANGLES • Find x and y. 18 x y

SPECIAL RIGHT TRIANGLES • Sometimes, the numbers won’t come out nice and neat. •

SPECIAL RIGHT TRIANGLES • Sometimes, the numbers won’t come out nice and neat. • For instance…say I had these dimensions. How would I find x and y? y x 40. 5

SPECIAL RIGHT TRIANGLES • Since the given side is the longer leg (l) all

SPECIAL RIGHT TRIANGLES • Since the given side is the longer leg (l) all we do to find x is remember the relationship s √ 3 = 40. 5 and solve for s (the short leg). • This equals… 23. 4 x y 40. 5

SPECIAL RIGHT TRIANGLES • From there, we can find y by simple doubling our

SPECIAL RIGHT TRIANGLES • From there, we can find y by simple doubling our s since the hypotenuse is 2 s. • Our y will equal… 46. 8 23. 4 y 40. 5

SPECIAL RIGHT TRIANGLES • To check our work we can do the Pythagorean Theorem.

SPECIAL RIGHT TRIANGLES • To check our work we can do the Pythagorean Theorem. • 23. 4 ² + 40. 5² = 2187. 81 • √ 2187. 81 = 46. 8 • PERFECT.

SPECIAL RIGHT TRIANGLES • Since all my 45 -45 -90 triangles and all my

SPECIAL RIGHT TRIANGLES • Since all my 45 -45 -90 triangles and all my 3060 -90 triangles have the same dimensions, this means that all triangles of like type will be similar. • All 45 -45 -90 s are similar to all other 45 -4590 s, and the same goes for 30 -60 -90 s.