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 median

Under the median of the triangle refers to the segment which connects one of the vertices of the triangle with the midpoint of the opposite side. The definition implies that every triangle has three medians.

You will need

- Knowledge of the lengths of the sides of the triangle.

Instruction

1

To calculate the length of the median formula is applied (see Fig. 1), where:

mc is the length of the median;

a, b, c be the sidelengths of a triangle.

mc is the length of the median;

a, b, c be the sidelengths of a triangle.

Note

The medians of a triangle have the following properties:

1) any of the three medians divides the original triangle into two equal size triangle;

2) All medians of a triangle have a common intersection point. This point is called the center of the triangle;

3) the Medians of a triangle divide it into 6 equal triangles. Equal are called geometric shapes with equal areas.

1) any of the three medians divides the original triangle into two equal size triangle;

2) All medians of a triangle have a common intersection point. This point is called the center of the triangle;

3) the Medians of a triangle divide it into 6 equal triangles. Equal are called geometric shapes with equal areas.

Useful advice

If the triangle is an isosceles triangle, its medians are equal. In addition, in such a triangle the medians coincide with the bisectors and heights.

The angle bisector is the ray that emanates from any vertex of a triangle and forming it divides the angle in half.

Under the height of a triangle is meant the segment, which is drawn from the vertex of the triangle perpendicular to the opposite side.

The angle bisector is the ray that emanates from any vertex of a triangle and forming it divides the angle in half.

Under the height of a triangle is meant the segment, which is drawn from the vertex of the triangle perpendicular to the opposite side.

# Advice 3 : 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 4 : How to find the median of a triangle

A median

**of a triangle**is the segment connecting any vertex**of a triangle**with the midpoint of the opposite side. The three medians intersect in one point always inside the**triangle**. This point divides each**median**in the ratio 2:1.Instruction

1

The median can be found using the theorem of Stewart. According to which, the square of the median is equal to a quarter of the amount of the matching squares of the sides minus the square of the side, which held the median.

mc^2 = (2a^2 + 2b^2 - c^2)/4,

where

a, b, c are sides

mc is the median to side C;

mc^2 = (2a^2 + 2b^2 - c^2)/4,

where

a, b, c are sides

**of a triangle**.mc is the median to side C;

2

The task of finding the median can be solved using additional constructions

2*(a^2 + b^2) = d1^2 + d2^2,

where

d1, d2 - diagonal of the resulting parallelogram;

from here:

d1 = 0.5*v(2*(a^2 + b^2) - d2^2)

**of the triangle**to the parallelogram and the solution using the theorem of the diagonals of a parallelogram.Extend the sides**of a triangle**and**the median**, to complete them to the parallelogram. Thus, median**of a triangle**is equal to half the diagonal of the resulting parallelogram, two sides**of the triangle**- its sides (a, b), and the third side**of the triangle**, which was a median, is the second diagonal of the resulting parallelogram. According to the theorem, the sum of the squares of the diagonals of a parallelogram is twice the sum of the squares of its sides.2*(a^2 + b^2) = d1^2 + d2^2,

where

d1, d2 - diagonal of the resulting parallelogram;

from here:

d1 = 0.5*v(2*(a^2 + b^2) - d2^2)

# Advice 5 : How to find the angle of a triangle if you know two sides?

In a right triangle, you simply find the angle if you know two sides of him. One of the angles is equal to 90 degrees, the other two are always sharp. These angles and will need to find. In order to find an acute angle in a right triangle, you must know the values of all three sides. Depending on which side you are aware, the sinuses acute angles can be found using formulas for trigonometric functions. To find the angle values for the sine uses a four digit mathematical table.

You will need

- - Pythagorean Theorem;
- - trigonometric function sin;
- - four-digit mathematical tables Bradis.

Instruction

1

Use the following notation for ease of preparation of the formulas necessary for calculations: c is the hypotenuse of a right triangle; a, b are the legs that form a right angle, A is the acute angle opposite sides of b; B is the acute angle opposite of side a.

2

Calculate what is the length of the unknown side of the triangle. Apply for computing the Pythagorean theorem. Calculate the side a, if the known values of the hypotenuse c and leg b. To do this, subtract the square of the hypotenuse c, the square of side b and then calculate the square root of the result.

3

Calculate side b, if the known values of the hypotenuse c and a leg. To do this, subtract the square of the hypotenuse c, the square of side a, and then calculate the square root of the result.

4

Calculate the value of the hypotenuse c, if there are two sides. This will get the sum of the squares of the legs a and b and then calculate the square root of the result and, if necessary, rounded to four decimal places.

5

Calculate the sine of the angle A according to the formula sinA = a/c. Use for calculations the calculator. Rounded, if necessary, the value of the sine of the angle A to four decimal places.

6

Calculate the sine of the angle B by the formula sinB = b/c. Use for calculations the calculator. Rounded, if necessary, the value of the sine of the angle B to four decimal places.

7

Find the angles A and B by the values of their sines. Use to determine the values of the angles of table VIII four-digit mathematical tables Bradis. Find in this table the values of sine. Move from values to the left and the first column "And" take a degree. Move from the found values up and from the top row "And" take minutes. For example, if sin(A) = 0,8949, the angle A is 63 degrees 30 minutes.

# Advice 6 : 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**.# Advice 7 : How to find the length of medians of the triangle

Median of a triangle is a segment drawn from any vertex to the opposite side, it divides it into parts of equal length. The maximum number of medians in a triangle is three, the number of vertices and sides.

Instruction

1

Task 1.

In an arbitrary triangle ABD held the median BE. Find its length if it is known that the parties, respectively, equal to AB = 10 cm, BD = 5 cm and AD = 8 cm.

In an arbitrary triangle ABD held the median BE. Find its length if it is known that the parties, respectively, equal to AB = 10 cm, BD = 5 cm and AD = 8 cm.

2

Solution.

Apply the formula median expression across all sides of the triangle. This is a simple task, since all the lengths of the sides are known:

BE = √((2*AB^2 + 2*BD^2 - AD^2)/4) = √((200 + 50 - 64)/4) = √(46,5) ≈ 6,8 (cm).

Apply the formula median expression across all sides of the triangle. This is a simple task, since all the lengths of the sides are known:

BE = √((2*AB^2 + 2*BD^2 - AD^2)/4) = √((200 + 50 - 64)/4) = √(46,5) ≈ 6,8 (cm).

3

Task 2.

In the isosceles triangle ABD, the sides AD and BD are equal. Held the median from vertex D to the side of the BA, it is BA with the angle equal to 90°. Find the length of the median DH, if you know that BA = 10 cm, and the angle DBA equal to 60°.

In the isosceles triangle ABD, the sides AD and BD are equal. Held the median from vertex D to the side of the BA, it is BA with the angle equal to 90°. Find the length of the median DH, if you know that BA = 10 cm, and the angle DBA equal to 60°.

4

Solution.

To find the median, identify one and equal sides of the triangle AD or BD. For this we consider one of the right triangles, suppose BDH. From the definition of the median, it follows that BH = BA/2 = 10/2 = 5.

Find the side BD in the formula of the trigonometric properties of a right triangle - BD = BH/sin(DBH) = 5/sin60° = 5/(√3/2) ≈ 5,8.

To find the median, identify one and equal sides of the triangle AD or BD. For this we consider one of the right triangles, suppose BDH. From the definition of the median, it follows that BH = BA/2 = 10/2 = 5.

Find the side BD in the formula of the trigonometric properties of a right triangle - BD = BH/sin(DBH) = 5/sin60° = 5/(√3/2) ≈ 5,8.

5

Now there are two options for finding the median: the formula used in the first task or the Pythagorean theorem for a right triangle BDH: DH^2 = BD^2 - BH^2.

DH^2 = (5,8)^2 - 25 ≈ 8,6 (cm).

DH^2 = (5,8)^2 - 25 ≈ 8,6 (cm).

6

Task 3.

In an arbitrary triangle BDA carried out the three medians. Find the length if it is known that the height of the DK equal to 4 cm and divides the base into segments of length BK = 3 and KA = 6.

In an arbitrary triangle BDA carried out the three medians. Find the length if it is known that the height of the DK equal to 4 cm and divides the base into segments of length BK = 3 and KA = 6.

7

Solution.

To find the median of the necessary lengths of all sides. The length of the BA can be found from the conditions: BA = BH + HA = 3 + 6 = 9.

Consider a right triangle BDK. By the Pythagorean theorem find the length of the hypotenuse BD:

BD^2 = BK^2 + DK^2; BD = √(9 + 16) = √25 = 5.

To find the median of the necessary lengths of all sides. The length of the BA can be found from the conditions: BA = BH + HA = 3 + 6 = 9.

Consider a right triangle BDK. By the Pythagorean theorem find the length of the hypotenuse BD:

BD^2 = BK^2 + DK^2; BD = √(9 + 16) = √25 = 5.

8

Similarly, find the hypotenuse of a right triangle KDA:

AD^2 = DK^2 + KA^2; AD = √(16 + 36) = √52 ≈ 7,2.

AD^2 = DK^2 + KA^2; AD = √(16 + 36) = √52 ≈ 7,2.

9

According to the formula expressions using side find the median:

BE^2 = (2*BD^2 + 2*BA^2 - AD^2)/4 = (50 + 162 - 51,8)/4 ≈ 40, hence BE ≈ 6,3 (cm).

DH^2 = (2*BD^2 + 2*AD^2 - BA^2)/4 = (50 + 103,7 - 81)/4 ≈ 18,2, hence DH ≈ 4,3 (cm).

AF^2 = (2*AD^2 + 2*BA^2 - BD^2)/4 = (103,7 + 162 - 25)/4 ≈ 60, hence AF ≈ 7,8 (cm).

BE^2 = (2*BD^2 + 2*BA^2 - AD^2)/4 = (50 + 162 - 51,8)/4 ≈ 40, hence BE ≈ 6,3 (cm).

DH^2 = (2*BD^2 + 2*AD^2 - BA^2)/4 = (50 + 103,7 - 81)/4 ≈ 18,2, hence DH ≈ 4,3 (cm).

AF^2 = (2*AD^2 + 2*BA^2 - BD^2)/4 = (103,7 + 162 - 25)/4 ≈ 60, hence AF ≈ 7,8 (cm).

# Advice 8 : How to calculate side of isosceles triangle

Isosceles, or isosceles called triangle whose lengths of two sides of the same. If necessary, calculate the length of one of the sides of such figures it is possible to use knowledge of angles at its vertices, in combination with the length of one side or radius of the circumscribed circle. These parameters of the polygon are linked by a theorem of sines, cosines, and some other permanent ratio.

Instruction

1

To calculate the length of the sides of an isosceles triangle (b) according to the known conditions of the length of its base (a) and the value of the adjacent angle (α) use the theorem of cosines. It implies that you should divide the length of the known sides for twice the cosine is given in terms of angle: b = a/(2*cos(α)).

2

The same theorem and apply for the reverse operation - calculate the length of the base (a) at a known length of sides (b) and value of angle (α) between these two parties. In this case, the theorem allows to obtain the equality, the right part of which contains twice the product of the lengths of the known sides into the cosine of the angle: a = 2*b*cos(α).

3

If in addition the lengths of the sides (b) in terms of the magnitude of the angle between them (β), to calculate the length of the base (a) use the theorem of sines. It implies a formula, according to which should be twice the length of the side multiplied by the sine of half of the known angle: a = 2*b*sin(β /2).

4

The theorem of sines can be used to find the length of the lateral side (b) of an isosceles triangle if the known base length (a) and opposite him the value of angle (β). In this case, double the sine of half of the known angle divided by the resulting value of the length of the base: b = a/(2*sin(β/2)).

5

If the isosceles triangle circumscribed circle, the radius of which (R) is known, to calculate the lengths of the sides need to know the measure of the angle at one vertex of the shape. If the conditions given information about the angle between the sides (β), calculate the length of the base (a) of the polygon by doubling the product of the radius by the sine of this angle: a = 2*R*sin(β). If given the value of the base angle (α) to find the length of the lateral side (b) just replace the angle in the formula: b = 2*R*sin(α).