Pythagorean Theorem and Special Right Triangles

Pythagorean Theorem and Special Right TrianglesGallegoGarridosLopozRadjacTejano

Pythagorean Theorem

DefinitionIn a right angled triangle:the square of the hypotenuse is equal to the sum of the squares of the other two sides.

If ABC is a right triangle, thena2+ b2= c2

hypotenuselegleg

ACB

Why Is This Useful?If we know the lengths oftwo sidesof a right angled triangle, we can find the length of thethird side.

(But remember it only works on right angled triangles!)

Equationa2+ b2= c2

Example #1

Using the Pythagorean theorem, does this triangle have a right angle?

Example #1 Solutiona2+ b2= c2?c2= a2+ b2c2= 102+ 242c2= 100 + 576 c2= 676c2= 262=676They are equal, so ...Yes, it does have a right angle!

Example #2Find the length of the hypotenuse.

16 in30 inx

Example #2 SolutionPythagorean Theorem; a2+ b2= x2?x2 = a2+ b2x2 = 302 in + 162 inx2 = 900 in2 + 256 in2x2 =1156 in2x=34 in

16 in30 inx

Pythagorean TriplesA Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation a2+ b2= c2 .Common Pythagorean Triples and Some of their Multiples

3, 4, 55, 12, 138, 15, 177, 24, 256, 8, 1010, 24, 2616, 30, 3414, 48, 5030, 40, 5050, 120, 13080, 150, 17070, 240, 2503x, 4x, 5x5x, 12x, 13x8x, 15x, 17x7x, 24x, 25x

The most common Pythagorean triples are in bold. The other triples are the result of multiplying each integer in a bold face triple by the same factor.

Example #1Based on these Pythagorean triples:

Find the length of the third side and tell whether it is a leg or the hypotenuse.

3, 4, 55, 12, 138, 15, 177, 24, 2520, 5224, 4521, 7518, 3080, 150

Example #1Solution

1. 20, 48

2. 24, 454. 18, 30

3. 21, 755. 80, 150

A common Pythagorean triple is 5,12,13. Notice that if you multiply the lengths of the legs of the Pythagorean triple by 4, you get the lengths of the legs of the triangle: 5(4) = 20 and 12(4) = 48 and the hypotenuse is 13(4) = 52.

To complete the Pythagorean triple, the third side is 52 and is the hypotenuse.

(The same method for the rest)

Third side : 51Hypotenuse

Third side : 72LegThird side : 170HypotenuseThird side : 24Leg

Special Right Triangles The "special" nature of these triangles is their ability to yield exact answers instead of decimal approximations when dealing with trigonometric functions.

Special Right Triangles 30- 60- 90 TheoremIn this theorem, the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg.

60302xx3x

Special Right Triangles 45- 45- 90 TheoremIn this theorem , the hypotenuse is 2 times as long as each leg.

4545x2xx

Example #1Find the value of x.

6045 mx

Example #2Find the value of x.

15 ft15 ftxx

PROOF OF PYTHAGOREAN THEOREMThere are many proofs of the Pythagorean Theorem. An informal/direct proof will be shown.

Proving

ccccaaaabbbb

ProvingThe four right triangles are congruent, and they form a small square in the middle.

Now, how can you solve for the area of the large square?

ccccaaaabbbb

ProvingThe area of the large square is equal to the area of the four triangles plus the area of the smaller square.

ccccaaaabbbb Area of large square

=+

Area of 4 triangles Area of small square

Proving(a + b) 2 = 4(1/2) (ab) + c2(a + b) 2 = 4(ab/2) + c2(a + b) 2 = 2ab + c2a2 + 2ab + b2 = 2ab + c2a2 + b2 = c2c2 = a2 + b2 Area of large square

=+

Area of 4 triangles Area of small square by area formulasDivisionMultiplicationSubtract 2ab on both sidesReflexive Property

CONVERSE OF PYTHAGOREAN THEOREMHow can you classify a triangle according to its side length when using pythagorean theorem and its given lengths?

Converse of Pythagorean TheoremIf the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengts of the other two sides, then the triangle is a right triangle.

If a2+ b2= c2 , thenABC is right triangle.longestside 2side1ACB

Converse of Pythagorean TheoremIf the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is an ACUTE TRIANGLE.

If c2 < a2+ b2 , thenABC is obtuse triangle.

ACB

Converse of Pythegorean TheoremIf the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is an OBTUSE TRIANGLE.

If c2 > a2+ b2 , thenABC is obtuse triangle.

ACB

PracticeClassify if its acute, obtuse, or right triangle with the given side length measurements.

1, 12, 16,9,103,7,413,6,122,4,5

Right Acute ObtuseAcuteObtuse

Sourceshttps://www.mathsisfun.com/pythagoras.htmlhttp://www.regentsprep.org/regents/math/algtrig/att2/ltri30.htmhttp://www.regentsprep.org/regents/math/algtrig/att2/ltri45.htm

QUIZ!Answer on a crosswise sheet of paper

#1Derive the value of x using the given measurements and Pythagorean Theorem.

x3f2d

#2Find the height h.

10 cm12 cmh

#3Find the length of the hypotenuse.

450

52 cm

#4Find the value of x.

60060060060015 mx

#5Find the value of y.

60018 fty