If you know that the triangle is rectangular, it gives you knowledge of the value of one of the corners, i.e., missing for calculation of the third parameter. The required side (C) hypotenuse - the side lying opposite the right angle. Then to calculate, extract square root and squared and folded the lengths of the other two sides (A and B) of this figure: C=√(A2+B2). If the desired direction is the leg, remove the square root from the difference between the squares of the lengths of the longer (hypotenuse) and lower (second leg) sides: C=√(A2-B2). These formulas are derived from the Pythagorean theorem.
Knowledge as a third parameter to the perimeter of the triangle (P) reduces the problem of computing the length of a missing side (S) to a simple subtraction operation - subtract from perimeter the length of both (A and B) famous faces: C=P-A-B. This formula follows from the definition of the perimeter which is the length of the polygonal line bounding the area of the shape.
The presence of baseline angle (γ) between the sides (A and B) of known length will require to find the length of the third (C) computation of trigonometric functions. Lift both the lengths of the sides in the square and add the results. Then, from this value subtract the product of their lengths into the cosine of the known angle, and in the end remove from the obtained values of the square root: C = √(A2+B2-A*B*cos(γ)). The theorem that you used in the calculations, is called the theorem of sines.
A well-known area of the triangle (S) will require the use of three formulas. The first defines the area as half of the work length of the known sides (A and B) the sine of the angle between them. Express from it the sine of an angle and you will get the expression 2*S/(A*B). The second formula allows to Express the cosine of the same angle as the sum of the squares of the sine and cosine of the same angle equals one, the cosine is the square root of the difference between the unit and the square received the previous expression: √(1-(2*S/(A*B))2). The third formula is a theorem, the law of cosines was used in the previous step, replace in it is the cosine of the resulting expression and you will have the following formula for calculation: C = √(A2+B2-A*B*√(1-(2*S/(A*B))2)).
Advice 2: How to find third side of triangle
A closed geometric figure with three angles of non-zero magnitude is called a triangle. Knowledge of the sizes of the two sides is not enough to calculate the length of the third side, we should know the value of at least one of the corners. Depending on the mutual arrangement of the known sides and angle for calculations should apply different methods.
If the conditions of the problem except for the lengths of two sides (A and C) in an arbitrary triangle is known and the angle between them (β), then apply to find the length of the third side (B) the theorem of cosines. First, lift the lengths of the sides into a square and fold the values obtained. From this value subtract twice the product of the lengths of these sides into the cosine of the known angle, and from what remains, remove the square root. In General, the formula can be written as: B=√(A2+C2-2*A*C*cos(β)).
If given the measure of an angle (α), which lies opposite the longer of (A) two known sides, then start with calculating the angle opposite the other known side (B). If we start from the theorem of sines, its value must be equal to arcsin(sin(α)*B/A), this means that the magnitude of the angle lying opposite the unknown side will be 180°-α-arcsin(sin(α)*B/A). Following to find the expected length of the same theorem of sines, multiply the length of the longest side to the sine of the angle is found and divide by the sine-known because of the problem of the angle: C=A*sin(α-arcsin(sin(α)*B/A))*sin(α).
If given the measure of an angle (α) adjacent to the unknown side length (C), and the other two parties have identical and known according to the problem dimensions (A), the formula will be much easier. Find twice the product of known length into the cosine of the known angle: C=2*A*cos(α).
If we consider a rectangular triangle and is known the length of its two sides (A and b), then find the length of the hypotenuse (C) use the Pythagorean theorem. Extract the square root of the sum of squared lengths of the known sides: C=√(A2+B2).
If in a right triangle are known the length of one of the legs (B) and the hypotenuse (C), to calculate the length of the other leg comes from the same theorem. Extract the square root of the difference between the built in the squares of the lengths of the hypotenuse and the known leg: C=√(C2-B2).
Advice 3: How to find the side of a triangle
Side of the triangle is a direct, limited its vertices. All of them have figures of three, this number determines the number of almost all graphics characteristics: angle, midpoint, bisectors, etc. to find the side of the triangle, you should carefully examine the initial conditions of the problem and determine which of them may be basic or intermediate values to calculate.
The sides of the triangle, like other polygons have their own names: the sides, base, and hypotenuse and legs of the figure with a right angle. This facilitates the calculations and formulas, making them more obvious even if the triangle is arbitrary. The figure of graphics, so it is always possible to arrange so as to make the solution of the problem more visible.
Sides of any triangle are connected and the other characteristics of the various ratios that help calculate the required value in one or more actions. Thus the more complex the task, the longer the sequence of steps.
The solution is simplified if the triangle is standard: the words "rectangular", "isosceles", "equilateral" immediately allocate a certain relationship between its sides and angles.
The lengths of the sides in a right triangle are connected by Pythagorean theorem: the sum of the squares of the legs equals the square of the hypotenuse. And the angles, in turn, are associated with the parties to the theorem of sines. It affirms the equality relations between the lengths of the sides and the trigonometric function sine of the opposite angle. However, this is true for any triangle.
Two sides of an isosceles triangle are equal. If their length is known, it is enough only one value to find the third. For example, suppose we know the height held to it. This cut divides the third side into two equal parts, and allocates two rectangular triangleH. Considered one of them, by the Pythagorean theorem find the leg and multiply it by 2. This will be the length of an unknown side.
Side of the triangle can be found through other sides, corners, length, altitude, median, bisector, perimeter size, area, inradius, etc. If you can't apply the same formula to produce a series of intermediate calculations.
Consider an example: find side of an arbitrary triangle, knowing the median ma=5, held for her, and the lengths of the other two medians mb=7 and mc=8.
Resented involves the use of formulas for the median. You need to find the way . Obviously, there should be three equations with three unknowns.
Write down the formulae for the medians:ma = 1/2•√(2•(b2 + c2) – a2) = 5;mb = 1/2•√(2•(a2 + c2) – b2) = 7;mc = 1/2•√(2•(a2 + b2) – c2) = 8.
Express c2 from the third equation and substitute it into the second:c2 = 256 – 2•a2 – 2•b2 b2 = 20 → c2 = 216 – a2.
Lift both sides of the first equation in the square and find a by entering explicit values:25 = 1/4•(2•20 + 2•(216 – a2) – a2) → a ≈ 11,1.