You will need

- a blank sheet of paper;
- pencil;
- line;
- textbook on geometry.

Instruction

1

Consider the right triangle ABC, where ∠ABC = 90°. Omit from this angle, the altitude h to the hypotenuse AC, the intersection of the altitude with the hypotenuse denoted by D.

2

Triangle ADB is similar to triangle ABC two angles are: ∠ABC = ∠ADB = 90°, ∠BAD is common. The similarity of triangles we obtain the ratio: AD/AB = BD/BC = AB/AC. Take the first and last ratio, proportion, and we find that AD = AB2/AC.

3

Since the triangle ADB is right angle, for his fair the Pythagorean theorem: AB2 = AD2 + BD2. Substitute in this equality AD. It turns out that BD2 = AB2 - (AB2/AC)2. Or what is the same, BD2 = AB2(AC2-AB2)/AC2. Because the rectangular triangle ABC, AC2 - AB2 = BC2, then get BD2 = AB2BC2/AC2 or, extracting the square root of both sides, BD = AB*BC/AC.

4

On the other hand, also the triangle BDC is similar to triangle ABC two angles are: ∠ABC = ∠BDC = 90°, ∠DCB - General. From the similarity of these triangles we get the aspect ratio of BD/AB = DC/BC = BC/AC. This proportion is expressed through the DC side of the original right triangle. To do this, consider the second equality in proportion and we find that DC = BC2/AC.

5

From the ratio obtained in step 2, have that AB2 = AD*AC. Step 4 have that BC2 = DC*AC. Then BD2 = (AB*BC/AC)2 = AD*AC*DC*AC/AC2 = AD*DC. Thus, the height of BD is the square root of the product of AD and DC, or as they say, the average geometric parts into which the altitude divides the hypotenuse of the triangle.

Note

If the altitude drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangle similar to the original and similar to each other.

Useful advice

In a right triangle, both sides act as the heights.

# Advice 2: How to find the hypotenuse in a right triangle

Call the hypotenuse side in a right triangle that lies opposite the right angle. The hypotenuse is the longest side in a right triangle. The other sides in a right triangle are called legs.

You will need

- Basic knowledge of geometry.

Instruction

1

The squared length of the hypotenuse equals the sum of the squares of the other two sides. That is, to find the square of the length of the hypotenuse you need to square the lengths of the legs and fold.

2

The length of the hypotenuse is equal to the square root of the square of its length. To find its length, extract the square root of the number equal to the sum of the squares of the legs. The resulting number will be the length of the hypotenuse.

Note

The length of the hypotenuse is a positive value, therefore, when extracting the root, radical expression must be greater than zero.

Useful advice

In an isosceles right triangle the length of the hypotenuse can be calculated by multiplying the leg to the root of the two.