Lessons

1. Basics
 2. Deductive Reasoning
3 - Parallel lines
4 - Congruent Triangles
5 - Quadrilaterals
6 - Inequalities
7 - Similar Polygon
8 - Rt. Triangles
9 - Circles
10 - Constructions
11 - Areas of 2D objects
12 - Areas and Volumes
13 - Coordinates

 

Right Triangles

 

Objective:

            • Learning about the geometric mean between two numbers, pythagorean theorem

          • Introduction to special right triangles

 

Lesson 8-1 Right Triangles:

            Geometric mean: If a, b, and X are positive numbers with a/X =X/b, then X is the geometric mean between a and b

                                    Example: 

     Find the geometric mean between 6 and 11

    Solution I (Using proportion)  6/x =x/11: x2=6(11); x= 8.124

Solution II (using the equation x=)= = 8.124

 

Theorem 8-1

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other

 Triangle ABC, triangle ADB, and triangle BDC are all similar

 

Corollary I

When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.

AD/BD=BD/CD

 

Corollary II

When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

AC/AB=AB/AD

AC/CB=CB/BC

 

Theorem 8-2 (Pythagorean Theorem)

 In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs

A2+B2=C2

Theorem 8-3

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle

 

Triangle ABC is a right triangle because 32+42=52

If C2 = A2+B2, then measure of angle C=90,and triangle ABC is right

Theorem 8-4

If C2 < A2+B2, then measure of angle C<90,and triangle ABC is acute

Theorem 8-5

If C2 > A2+B2, then measure of angle C>90,and triangle ABC is obtuse

Theorem 8-6

In a 45º-45º-90º, the hypotenuse is times as long as a leg.

 

Theorem 8-7

In a 30º-60º-90º triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg

 

 

 

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