Type of Class:
          Standard High School Geometry

Related VA SOL: G.7

Time Frame: Ninety-minute block period

Objectives:

• Students will be able to identify the special right triangles 45-45-90 and 30-60-90.

• Students will be able to create formulae to find lengths of the legs and hypotenuse of the special right triangles.

• Students will solve practical problems using special right triangles.

Materials: Chalkboard

Procedures:

1)      Word problem[1] to put things into context

a)      A baseball diamond is a square with sides of 90 feet long.

i)        To the nearest foot, what is the length of a throw from home to 2^nd base?

ii)       What is the length of a throw from the shortstop position to 1^st base if the short stop is midway between 2^nd and 3^rd base?

2)      Exploring 45-45-90 isosceles right triangles

a)      Questions directed to students

i)        What is an isosceles triangle?

ii)       How would you describe an isosceles triangle?

iii)     What is the angle measure of the two congruent angles?  How do you know?

iv)     Can you form a conjecture about an isosceles right triangle and its angle measurements?

Example of an isosceles triangle

v)      Who can remind the class of the Pythagorean Theorem?

(1)   A² + B² = C²

vi)     Can you find the length of ‘c’ using the Pythagorean Theorem for isosceles triangles?

      c = ?                                     c = ?                        c = ?

vii)   Can anyone form a conjecture between the length of the hypotenuse and the legs of an isosceles triangle?

b)      Given triangle ABC with a right angle at C: AC = BC = x, prove AB =  

i)        (AB)² = x² + x² = 2x²                                           

ii)       So AB =  =

c)      How would you find the length of the legs of an isosceles triangle given the hypotenuse?

d)      Students to solve the given problem in pairs, with one group coming to the board for their answer.

3)      Exploring 30-60-90 right triangles

a)      If triangle ABC is equilateral with altitude BD, what are the measures of: 

i)        This is the 2^nd special right triangle; 30-60-90

b)      Solve for x and y in the following examples with 30-60-90 right triangle            

c)      Can we form a conjecture about 30-60-90 triangles regarding their angles and side lengths?

i)        In a 30-60-90 right triangle, the measure of the hypotenuse is two times that of the leg opposite the 30-degree angle.  The measure of the other leg is  times that of the leg opposite the 30-degree angle.

4)      Back to the original word problem to be solved by students in pairs

a)      A baseball diamond is a square with sides of 90 feet long.

i)        To the nearest foot, what is the length of a throw from home to 2^nd base?

b)      What is the length of a throw from the shortstop position to 1^st base if the short stop is midway between 2^nd and 3^rd base?

Assessment: Complete the attached worksheet.

Suggestions/Comments: A worksheet comprised of a lot of 45-45-90 and 30-60-90 triangles in solving for angles and side measurements would be a good way for students to grow accustom to using the new properties.