Instruction

1

Suppose that we are given triangle ABC is isosceles. Known length its sides and base. Need to find the median, descended on the base of this triangle. In an isosceles triangle the median is also the median, bisector and height. Due to this property, find the median to the base of the triangle is very simple. Use Pythagorean theorem for the right triangle ABD: AB2 = BD2 + AD2, where BD is the desired median, AB - side (for convenience, let it equal a), and AD is half of the base (for convenience, take the basis equal to b). Then BD2 = a2 - b2/4. Find the root of this expression and get the length of the median.

2

Slightly more complicated is the situation with the median drawn to the lateral side. To start, draw both of these medians in the figure. These medians are equal. Label the side with the letter a, and the base is b. Mark equal angles at the base α. Each median divides the side into two equal parts a/2. Mark the desired length of the median x.

3

Cosine theorem to Express any side of a triangle using the other two and the cosine of the angle between them. We can write the theorem of cosines for the triangle AEC: AE2 = AC2 + CE2 - 2AC·CE·cos∠ACE. Or what is the same, (3x)2 = (a/2)2 + b2 - 2·ab/2·cosα = a2/4 + b2 - ab·cosα. For this task the well-known, but the angles at the base there, so the calculation continues.

4

Now apply the theorem of cosines to triangle ABC to find the angle at the base: AB2 = AC2 + BC2 - 2AC·BC·cos∠ACB. In other words, a2 = a2 + b2 - 2ab·cosα. Then cosα = b/(2a). Substitute this expression into the previous one: x2 = a2/4 + b2 - ab·cosα = a2/4 + b2 - ab·b/(2a) = a2/4 + b2 - b2/2 = (a2+2b2)/4. Calculating the root of the right side of the expression, you'll find the median drawn to the lateral side.

# Advice 2: How to find the side of isosceles triangle

Isosceles is called a triangle whose 2 sides are equal. From the definition it follows that a right triangle is also isosceles, but the converse is wrong. There are several ways to calculate the sides of an isosceles

**triangle**.You will need

- Know the angles of a triangle and at least one of its sides.

Instruction

1

Method 1. Out of the theorem of sines of a triangle. The theorem of sines States that: the sides of the triangle are proportional to the sines of the opposite angles (Fig. 1)

From this formula implies the following equality:a = 2Rsinα,b = 2Rsinβ

From this formula implies the following equality:a = 2Rsinα,b = 2Rsinβ

2

Method 2. Out of the cosines of the triangle. According to this theorem, for any plane triangle with sides a, b, c and angle α, which lies opposite the side, true equality in Fig. 2

Hence, there is a consequence:a = b/2cosα;

Also from the theorem of cosines, there is 1 more result:

b = 2a*sin(β/2)

Hence, there is a consequence:a = b/2cosα;

Also from the theorem of cosines, there is 1 more result:

b = 2a*sin(β/2)

# Advice 3: How to find the median of a number

For the generalized assessment of long -

**range**values are different helper methods and variables. One of these values is the median. Although it can be called the average value**of a number**, but its meaning and the method of its calculation differ from other variations on the theme of average.Instruction

1

The most common method to measure an average value in the range of values is the arithmetic mean. In order to calculate it, you need the sum of all values

**of a series**divided by the number of these values. For example, if given a number 3, 4, 8, 12, 17, its arithmetic mean is equal to (3 + 4 + 8 + 12 + 17)/5 = 44/5 = 8,6.2

Another average commonly encountered in mathematical and statistical tasks, called harmonic mean. Harmonic mean of the numbers a0, a1, a2... an is equal to n/(1/a0 + 1/a1 + 1/a2... +1/an). For example, the same

**number**as in the previous example, the harmonic mean will be equal 5/(1/3 + 1/4 + 1/8 + 1/12 + 1/17) = 5/(347/408) = 5,87. Harmonic mean is always less than the arithmetic mean.3

Different medium are used in different types of tasks. For example, if you know that the first hour the car was traveling at a speed A, and the second — with speeds of B, then its average speed during the journey will be the average between A and B. But if you know that the car traveled one kilometer at a speed A, and the next with the speed B, then to calculate his average speed during the journey, you will need to take the harmonic mean between A and B.

4

For statistical purposes, the arithmetic mean is a convenient and objective evaluation, but only in those cases when the values

**of a number**not dramatically eye-catching. For example,**the number of**1, 2, 3, 4, 5, 6, 7, 8, 9, 200 the arithmetic mean is equal to 24, 5 — much more of all the members**of the series**except the last one. It is obvious that such an estimate cannot be considered fully adequate.5

In such cases, you must calculate

**the median****of a number**. This average value, the value of which is exactly in the middle**of the row**so that all members**of the series**located before the median — not more than it, and all that comes after — not less. Of course, you need to first arrange the members**of a number**in ascending order.6

If the row a0,... an odd number of values, i.e. n = 2k + 1, then

For example, already considered a number of 1, 2, 3, 4, 5, 6, 7, 8, 9, 200 ten members. Consequently, the median is the arithmetic mean between the fifth and sixth members, that is (5 + 6)/2 = 5,5. This estimate is much better reflects the average value of a typical member of the

**the median**is taken by a member**of a number of**with a serial number k + 1. If the number of values is even, i.e. n = 2k, then median is considered to be the arithmetic mean of the members**of a number of**numbers k and k + 1.For example, already considered a number of 1, 2, 3, 4, 5, 6, 7, 8, 9, 200 ten members. Consequently, the median is the arithmetic mean between the fifth and sixth members, that is (5 + 6)/2 = 5,5. This estimate is much better reflects the average value of a typical member of the

**series**.