Lessons

1. Basics
2. Deductive Reasoning
3 - Parallel lines
4 - Congruent Triangles
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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