APPLYING RIGHT TRIANGLES AND TRIGONOMETRY

OBJECTIVE: SWBAT… FIND THE GEOMETRIC MEAN BETWEEN 2 NUMBERS SOLVE PROBLEMS INVOLVING RELATIONSHIPS BETWEEN PARTS OF A TRIANGLE AND THE ALTITUDE TO ITS HYPOTENUSE USE THE PYTHAGOREAN THEOREM AND ITS CONVERSE

GEOMETRIC MEAN: ax = xb EX: FIND THE GEOMETRIC MEAN BETWEEN 2 AND 10 EX: FIND THE GEOMETRIC MEAN BETWEEN 5 AND 25

EX: TO FIND THE HEIGHT OF HER SCHOOL BUILDING, ANN HELD A BOOK NEAR HER EYE SO THAT THE TOP AND BOTTOM OF THE BUILDING WERE WITH THE EDGES OF THE COVER. If ANN’S EYE LEVEL IS 5 FEET OFF THE GROUND AND SHE IS STANDING ABOUT 10 FEET FROM THE BUILDING, HOW TALL IS THE BUILDING? ASSUME THE BUILDING IS PERPENDICULAR TO THE GROUND AND THE EDGES OF THE COVER OF THE BOOK FORM RIGHT ANGLES.

THEOREM 8-1: IF THE ALTITUDE IS DRAWN FROM THE VERTEX OF THE RIGHT ANGLE OF A RIGHT TRIANGLE TO ITS HYPOTENUSE, THEN THE TWO TRIANGLES FORMED ARE SIMILAR TO THE GIVEN TRIANGLE AND TO EACH OTHER.

THEOREM 8-2: THE MEASURE OF THE ALTITUDE DRAWN FROM THE VERTEX OF THE RIGHT ANGLE OF A RIGHT TRIANGLE TO ITS HYPOTENUSE IS THE GEOMETRIC MEAN BETWEEN THE MEASURES OF THE TWO SEGMENTS OF THE HYPOTENUSE

THEOREM 8-3: IF THE ALTITUDE IS DRAWN TO THE HYPOTENUSE OF A RIGHT TRIANGLE, THEN THE MEASURE OF A LEG OF THE TRIANGLE IS THE GEOMETRIC MEAN BETWEEN THE MEASURES OF THE HYPOTENUSE AND THE SEGMENT OF THE HYPOTENUSE ADJACENT TO THAT LEG.

THEOREM 8-4: PYTHAGOREAN THEOREM IN A RIGHT TRIANGLE, THE SUM OF THE SQUARES OF THE MEASURES OF THE LEGS EQUALS THE SQUARE OF THE MEASURE OF THE HYPOTENUSE. a 2 + b 2 = c 2