The right triangle altitude theorem - practice problems

Euclid was a Greek mathematician and philosopher. He left us with two important but simple theorems that apply in a right triangle.

Euclid's first theorem (about height): The area of the square constructed above the height of the right triangle (h) is equal to the area of the rectangle constructed from both sections of the hypotenuse (c1 and c2):

h^{2} = c_{1} c_{2}

Or: The height in a right triangle is the geometric mean of two sections of the hypotenuse.

h = c_{1} \cdot c_{2}

Euclid's second theorem - about the hypotenuse: The area of the square constructed above the hypotenuse of a right-angled triangle is equal to the area of the rectangle constructed from the hypotenuse and the segment of the hypotenuse adjacent to this hypotenuse.

a^{2} = c \cdot c_{1}

b^{2} = c \cdot c_{2}

Or: The hypotenuse of a right triangle is the geometric diameter of the hypotenuse and the adjacent section of the hypotenuse.

a = c \cdot c_{1}

It is usually taught in high school. The Pythagorean theorem can be easily proved using Euclid's theorems.

Direction: Solve each problem carefully and show your solution in each item.