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 height when you know the length and width

At the base of many geometric shapes are rectangles and squares. The most common among them is a box. Also they include the cube, the pyramid and truncated pyramid. All these four figures have a parameter called height.

Instruction

1

Draw a simple isometric shape called a rectangular parallelepiped. It got its name for the reason that its faces are rectangles. The base of this parallelepiped is also a rectangle with width a and length b.

2

The volume of the rectangular prism equals the area of the base to

**the height**: V = S*h. Since the base of the box is the rectangle, the area of this base is equal to S=a*b, where a is**length**, b -*width*. Hence the volume is V=a*b*h, where h is the height (h = c, where c is the edge of the box). If the problem is to find**the height**of the box, convert the last formula as follows: h=V/a*b.3

There are rectangular parallelepipeds, in the bases of which are squares. All its faces are rectangles from which squares are two. This means that its volume is V=h*a^2, where h is the height of the parallelepiped, a -

**length**of a square equals the width. Accordingly,**the height**of this figure locate the following way: h=V/a^2.4

The cube has squares with the same parameters are all six faces. The formula for calculating volume is: V=a^3. To calculate any of the parties, if known to the other, is not required, since they are all equal.

5

All of the above methods involve the calculation of the height using the volume of the parallelepiped. However, there is another way to calculate

**the height**for a given width and length. It is used in that case, if the problem is the volume of the given square. The area of the parallelepiped is equal to S=2*a^2*b^2*c^2. Hence, c (height of the parallelepiped) is equal to C=sqrt(s/(2*a^2*b^2)).6

There are other tasks to calculate the height given the length and width. Some of these are the pyramids. If the problem you are given an angle with the plane of the base of the pyramid, and its

**length**and*width*, find**the height**using the Pythagorean theorem and properties of angles.7

In order to find

**the height**of the pyramid, first define the diagonal of the base. From the drawing it can be concluded that the diagonal is d=√a^2+b^2. Since the height falls in the center of the base, half of the diagonal of the find as follows: d/2=√a^2+b^2/2. Height find, using the properties of tangent: tgα=h/√(a^2+b^2/2. It follows that the height is h=√a^2+b^2/2*tgα.