The base of the triangle is the third side of the AC (see figure), perhaps distinct from side-points of equal sides AB and BC. Here are a few ways of calculating the base length of an isosceles triangle. First, you can use the theorem of sines. It States that the sides of the triangle is directly proportional to the value of the sines of the opposite angles: a / sin α = c / sin β. How do we obtain that c = a * sin β / sin α.
Here is an example of the calculation of the base of the triangle by the theorem of sines. Let a = b = 5, α = 30°. Then, by the theorem about sum of angles of a triangle β = 180° - 2 * 30° = 120°. C = 5 * sin 120° / sin 30° = 5 * sin 60° / sin 30° = 5 * √3 * 2 / 2 = 5 * √3. Here to compute the value of sine of an angle β = 120° we have used the formula of the coercion, according to which sin (180° - α) = sin α.
The second way to find the base of the triangle – using the spherical law of cosines: the square of the sides of a triangle is equal to the sum of the squares of the other two sides minus twice the product of these sides and the cosine of the angle between them. We find that the base square c^2 = a^2 + b^2 – 2 * a * b * cos β. Then we find the length of the base c, extracting the square root of this expression.
Let's consider an example. Suppose we are given the same parameters as in the previous task (see paragraph 2). a = b = 5, α = 30°. β = 120°. with^2 = 25 + 25 - 2 * 25 * cos 120° = 50 - 50 * (- cos 60°) = 50 + 50 * ½ = 75. In this calculation we applied the formula of bringing to find cos 120°: cos (180° - α) = - cos α. Extracted the square root and obtain the value of c = 5 * √3.
Consider a special case of an isosceles triangle – an isosceles right-angled triangle. Then by the Pythagorean theorem we immediately find a basis c = √(a^2 + b^2).
When evaluating an easy mistake to make in the values of the sine or cosine of the angle, or just in arithmetic operations. To check the result, it is useful to calculate the length of the base in two ways.
In calculating the angle opposite to the base, it will be convenient to use the following formulas casting: sin (180° - α) = sin α; cos (180° - α) = - cos α.
Advice 2 : How to calculate the angle of the triangle
Tregon define its corners and sides. The type of corners of such a trianglea squareand a sharp – all three angles acute obtuse one angle obtuse, right angle – one angle of a straight line, an equilateral triangle,a square, e all the angles equal 60. To find the angle, frictionangleand in different ways depending on the source data.
You will need
- basic knowledge of trigonometry and geometry
Calculate the angle, frictionangleand, if known to the other two angle α and β, as the difference between 180°−(α+β) as the sum of the angles ina square, e is always equal to 180°. For example, suppose there are two angle frictionangleand α=64°, β=45°, then the unknown angle γ=180−(64+45)=71°.
Use the theorem of cosines when you know the lengths of two sides a and b frictionanglea and angle α between them. Find the third side by the formula c=√(a2+b2−2*a*b*cos(α)), as the squared length of any side of trianglea squareand is equal to the sum of the squares of the lengths of the other sides minus twice the product of the lengths of these sides into the cosine of the angle between them. Write down the theorem of cosines for the other two sides: a2=b2+c2−2*b*c*cos(β), b2=a2+c2−2*a*c*cos(γ). Express of these formulas, the unknown angles: β=arccos((b2+c2−a2)/(2*b*c)), γ=arccos((a2+c2−b2)/(2*a*c)). For example, suppose tregone well-known side a=59, b=27, the angle between them α=47°. Then the unknown side c=√(592+272-2*59*27*cos(47°))≈45. Then β=arccos((272+452-592)/(2*27*45))≈107°, γ=arccos((592+452-272)/(2*59*45))≈26°.
Find the angles of trianglea squareand, if you know the lengths of all three sides a, b and c tregon. To do this, calculate the area of triangleangleand by Heron's formula: S=√(p*(p−a)*(p−b)*(p−c)), where p=(a+b+c)/2 – properiter. On the other hand, since the area of triangleangleand is equal to S=0,5*a*b*sin(α), we Express from this formula the angle α=arcsin(2*S/(a*b)). Similarly, β=arcsin(2*S/(b*c)), γ=arcsin(2*S/(a*c)). For example, suppose that we are given trea square with sides a=25, b=23 and C=32. Then count properiter p=(25+23+32)/2=40. Calculate the area by Heron's formula: S=√(40*(40-25)*(40-23)*(40-32))=√(40*15*17*8)=√(81600)≈286. Find the angles: α=arcsin(2*286/(25*23))≈84°, β=arcsin(2*286/(23*32))≈51°, and the angle γ=180−(84+51)=45°.
Advice 3 : How to calculate side of triangle
The triangle as a flat geometric figure made up of three parties that form the connection points (vertices of the) three angles. These angles and sides are connected by permanent relations, which allows one to find unknown side lengths when you have at least a minimal set of data about values of angles and lengths of other sides. Below are several ways to determine the length of sides of triangle in relation to the Euclidean plane.
If the known values of the two angles (α and β) and the length of one side (C), the lengths of the other two sides can be determined, but the calculation formula will vary, depending on, adjacent both known angle to the side of known length. If Yes, then based on the theorem of sines and given the theorem about sum of angles in a triangle, the length of the side (A), which lies opposite the angle α, can be defined as the ratio of product of the sine of that angle on a known side length to the sine of the difference between the unfolded angle (180°) and the sum of the two known angles: A=sin(α)∗C/(sin(180°-α-β)). To determine the length of the third side (B) lying opposite the angle β, this formula should be changed accordingly: B=sin(β)∗C/(sin(180°-α-β)).
If a party (B) of known length is not between the two known angles (α and β), and is adjacent to only one of them (e.g. α), the calculation formula of the lengths of the remaining sides will change. Side (C), lying opposite the unknown angle will have a length determined by the ratio of the product of the sine of the angle of losses to the total value of all angles is 180°, the length of the known side to the sine of the angle lying opposite to it: C=sin(180°-α-β)∗B/sin(β). And the length of the third side (A) can be determined by this formula: A=sin(α)∗B/sin(β).
If you know the lengths of two sides (A and B) and the angle of one corner, then to find the length of a missing side you can use the theorem of cosines. If the angle of known value (γ) lies between the known sides, the length of the required side (C) is equal to the square root of the difference between the sum of the squares of the lengths of the known sides and twice the product of the lengths of these sides into the cosine of the known angle: C=√(A2+B2-2∗A∗B∗cos(γ)).
Advice 4 : How to find the side of isosceles triangle if the base
A basic property of an isosceles triangle is equal to two adjacent sides and corresponding angles. You can easily find side of an isosceles triangleif the base and at least one element.
Depending on the circumstances of a particular task, you can find the side of an isosceles triangleif the base and any additional element.
The base and height to it.The perpendicular drawn to the base of an isosceles triangleis the concurrent height, median and bisector of the opposite angle. This interesting feature you can use by applying the theorem of Pythagoras:a = √(h2 + (c/2)2), where a is the length of the equal sides of the triangle, h is the height drawn to the base S.
The base and the altitude to one side.Drawing an altitude to the side, you'll get a rectangular triangle. The hypotenuse of one of them – the unknown side of an isosceles triangle, leg of a given height h. The second leg is unknown, label it H.
Consider the second triangle. Its hypotenuse is the basis of the total figures, one of the legs is equal to h. The other leg is the difference a – x. By the Pythagorean theorem, write down two equations relative to the unknown a and x:A2 = x2 + h2;c2 = (a - x)2 + h2.
Let the base equals 10 and height 8, then:A2 = x2 + 64;100 = (a - x)2 + 64.
Express artificially introduced variable x from the second equation and substitute it in the first and x = 6 → x = a – 6а2 = (a - 6)2 + 64 → a = 25/3.
The base and one of the equal angles α.Draw a height to the ground, consider one of the right triangles. The cosine of the lateral angle is equal to the ratio of the adjacent leg to the hypotenuse. In this case, the leg equal to half the base of an isosceles triangle, and the hypotenuse is the side:(c/2)/a = cos α → a = c/(2•cos α).
The base and opposite angle β.Drop a perpendicular on the base. A corner of one of the resulting right triangles is equal to β/2. The sine of that angle is the ratio of the opposite leg to the hypotenuse a, where:a = c/(2•sin(β/2))
Advice 5 : 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.
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)).
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).
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).
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)).
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).