8.1

Right Triangle Similarity Theorem

• If an altitude is drawn to the hypotenuse of a right triangle, the resulting triangles are similar to the original triangle and to each other.

  Example: In triangle ΔABD, ΔADB ∼ ΔBDC.

Geometric Mean

• Definition: The geometric mean of two positive numbers aa and bb is xx such that x2=abx2=ab or x=√(ab)x=√(ab).

• Example Calculations:

  • For 6 and 30: x=√(6×30)=6√5x=√(6×30)=6√5

  • For 16 and 28: x=√(16×28)=18√7x=√(16×28)=18√7

  • For 12 and 50: x=√(12×50)=10√6x=√(12×50)=10√6

Altitude Theorem

• The altitude length of a triangle is the geometric mean of the lengths of the two segments it divides the hypotenuse into: BD2=AD×DCBD2=AD×DC.

• Example Calculations:

  • For x and segments 12 and 8: x2=12×8x2=12×8 leading to x=4√6x=4√6.

Leg Theorem

• The length of a leg of the triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg:

  • For AB2=AD×ACAB2=AD×AC and BC2=DC×ACBC2=DC×AC.

• Example Calculations:

  • For a leg (x), hypotenuse 18, and adjacent segment 15: x2=15×18x2=15×18 leading to x=21.6x=21.6.

  • For yy in 212=30y212=30y: resulting in y=14.7y=14.7.