# The right triangle altitude theorem - practice problems - last page

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}⋅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.

$a_{2}=c⋅c_{1}$

$b_{2}=c⋅c_{2}$

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

$a=c⋅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.

#### Number of problems found: 66

- Touch circle

Point A has a distance (A, k) = 10 cm from a circle k with radius r = 4 cm and center S. Calculate: a) the distance of point A from the point of contact T if the tangent to the circle is drawn from point A b) the distance of the contact point T from the l - Hypotenuse - RT

A triangle has a hypotenuse of 55 and an altitude to the hypotenuse of 33. What is the area of the triangle? - The bridge

Across the circle, the lake passes through its center bridge over the lake. At three different locations on the lakeshore are three fishermen, A, B, and C. Which of the fishermen see the bridge from the largest angle? - Construct 61253

Using Euclid's theorem, construct a line of length √15.

- Same area

There is a given triangle. Construct a square of the same area. - Conical area

A right-angled triangle has sides a=12 and b=19 at the right angle. The hypotenuse is c. If the triangle rotates on the c side as an axis, find the volume and surface area of the conical area created by this rotation.

We apologize, but in this category are not a lot of examples.

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