Geometry 9.1 Similar Right Triangles:
I) Theorems:
A) 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.
B) In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.
The length of the altitude is the geometric mean of the lengths of the two segments.
C) In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
Geometry 9.2 The Pythagorean Theorem:
I) Theorems:
A) Pythagorean Theorem: in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
a2 + b2 = c2
Example: Given the two legs of a right triangle are 7 and 9, what is the hypotenuse?
72 + 92 = c2
49 + 81 = c2
130 = c2
c = √130 ≈ 11.40175425
B) Pythagorean Triple: is a set of three positive intergers, a, b, and c, that satisfy the equation
a2 + b2 = c2
Examples:
3, 4, 5
5, 12, 13
8, 15, 17
are a few
Geometry 9.3 The Converse of the Pythagorean Theorem:
I) Theorems:
A) Converse of the Pythagorean Theorem: if the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
If c2 = a2 + b2, then Δ ABC is a right triangle.
B) If 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 acute.
if c2 is less than a2 + b2, then ΔABC is acute.
C) If 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 obtuse.
if c2 is greater than a2 + b2, then ΔABC is obtuse.
( you can think of this in a doorway, the length of the door does not change nor does the doorway. As you open the door, the angle formed between the door and the doorway gets larger, therefore the distance from the doorway to the end of the door not on the hinge (in the room) is getting longer. So if the angle formed from the intersection of the doorway and door makes a right angle, the triangle is right. If the angle formed is smaller than the right angle, the triangle is acute. If the angle formed is larger than the right angle, the triangle is obtuse.)
Examples:
1) Given a triangle has lengths of 2, 10, and 11 is it acute, right or obtuse?
22 + 102 = 4 + 100 = 104
112 = 121
therefore the c2 is longer so the triangle is obtuse.
2) Given a triangle has lengths of 6, 5 and 7, is it acute, right or obtuse?
52 + 62 = 25 + 36 = 61
72 = 49
therefore the c2 is smaller so the triangle is acute.
3) Given a triangle has lengths of 6, 8 and 10, is it acute, right or obtuse?
62 + 82 = 36 + 64 = 100
102 = 100
therefore the c2 is equal so the triangle is right.
4) Given a triangle has lenghts of 5, 6, and 11, is it acute, right or obtuse?
Recall that a triangle has to have the sum of two sides greater than the third side and since 5 + 6 = 11, these side lenghts cannot form a triangle so it is none of them.
