Geometry: The Cathetus and Altitude Theorems

Vocabulary and theorems regarding right-angled triangles, including the Cathetus and Altitude Theorems, based on the provided lecture notes.

Hypotenuse (c)

The longest side of triangle $ABC$, which is opposite the right angle.

Catheti (a and b)

The two sides of the right-angled triangle $ABC$ that meet to form the right angle.

Similar Triangles

The triangles $ABC$, $ACH$, and $CBH$ which are similar because they each possess a right angle and share either angle $A$ or angle $B$.

Cathetus Theorem

In a right-angled triangle, the square over a cathetus is equal to the area of the rectangle formed by the hypotenuse and the associated hypotenuse segment ($a^2 = p \times c$ and $b^2 = q \times c$).

Hypotenuse Segment p

The section of the hypotenuse between point $A$ and the foot of the height $H$.

Hypotenuse Segment q

The section of the hypotenuse between point $B$ and the foot of the height $H$.

Segment Sum Identity

The relationship reflecting that the sum of the two hypotenuse segments equals the total hypotenuse: $q + p = c$.

Altitude Theorem (Höhensatz)

A geometric relationship used for proofs stating that the square of the height is equal to the product of the two hypotenuse segments: $h^2 = p \times q$.

Pythagorean Theorem

The formula used and reshaped for these geometric proofs: $a^2 + b^2 = c^2$.

Pythagorean relation for cathetus a

The relation representing the right triangle formed by cathetus $a$, altitude $h$, and segment $p$: $h^2 + p^2 = a^2$.

Pythagorean relation for cathetus b

The relation representing the right triangle formed by cathetus $b$, altitude $h$, and segment $q$: $h^2 + q^2 = b^2$.