Euclid's theorems - 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.
$h=\sqrt{c_1\cdot c_2}$

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.

$a=\sqrt{c\cdot c_1}$

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

Number of problems found: 61

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See also information about Euclid's theorems on Wikipedia.