diagonal/em/b" class="colorbox imagefield imagefield-imagelink" rel="gallery-step-images"> Rectangular quadrilaterals (rectangle, square) the diagonal divides into two right-angled triangles, in each of which she will be the hypotenuse. Therefore, for calculating it is possible to use the Pythagorean theorem. a2=b2+c2, where a is the hypotenuse, b and C are the legs. Example 1: find the diagonal AC , if you know that length BC=3 cm, AB=5 cm Solution: calculate the hypotenuse AC in right triangle ABC. AC2=AB2+BC2; AC2=52+32=34; from this value, extract the square root of a: AC=√34=5.8 to see the Answer: the diagonal of the rectangle is equal to 5.8 cm
If you see a square, then calculate the diagonal is possible, knowing one of its sides or the square. Since all the sides of the square are equal, the Pythagorean theorem is: a2=b2+b2, a2=2b2. Square - the product of two sides (S=b2). Hence, the square of the hypotenuse (in the figure the square) equal to twice the area (a2=2S).Example 2: the area of a square is 16 cm2. Find the length of the diagonal. Solution: calculate the length of the diagonal and through the area. a2=2S, a2=2*16 cm2=32; extract the square root a=√32≈5.7 cm answer: the length of the diagonal of the square – 5.7 cm
In some cases, to compute the diagonals necessary to make additional construction.Example 3: an equilateral polygon with sides equal to 6 cm, angle BCD straight. Find the length of diagonal AB.Solution: connect points B and D. it Turned out a right triangle BCD in which side BD is the hypotenuse. Calculate the hypotenuse BD: BD2=BC+CD2; BD2=62+62=72; the Hypotenuse BD of the triangle BCD is the leg of the triangle ABD. And the diagonal AB is the hypotenuse in it. Calculate diagonal AB: AB2=BD2+AD2=72+36=108; AB=√108=10,4 cm a: length of diagonal AB=10,4 cm
The diagonal of a cube can be found using the diagonal of one of its faces.Example 4: a cube with side 5 cm Find the diagonal of the cube.Solution: extend and calculate the diagonal of the cube face. AC2=52+52=50. The diagonal AC is perpendicular to the edge of the CB, therefore the angle ACB is a straight line. Diagonal cube AB is the hypotenuse of the triangle ACB. Find the length of diagonal of cube: AB2=AC2+CB2=50+25=75; extract the square root. AB=√75=8,7 cm answer: the length of the diagonal of a cube is 8.7 cm
To calculate the diagonals of a parallelogram using the theorem of cosines: c2=a2+b2-2ab*cosy.Example 5: a = 2 cm b = 3 cm, γ = 120°. Find the diagonal of s. Solution: substitute the values into the formula. c2= 22+32-2*2*3*cos120°; cos120° look at the table of cosines at (-0.5). S2 = 4+9-12*(-0,5)=13-(-6)=19. From this value, extract the root: C = √19 = 4,35 see Answer: length of diagonal = 4,35 cm
Advice 2: How to calculate the length of triangle side
To calculate lengths of sides in an arbitrary triangle most often has to apply the theorem of sines and cosines. But among the whole set of arbitrary polygons of this kind there are of them "more correct" variations - equilateral, isosceles, rectangular. If the triangle is known that it belongs to one of these species, methods of calculation of its parameters is much easier. When calculating the lengths of their sides is often possible to do without trigonometric functions.
The length of the side (A) of an equilateral triangle to find the radius of the inscribed circle (r). To do so, increase it to six times and divide by the square root of triples: A = r*6/√3.
Knowing the radius of the circumscribed circle (R), too it is possible to calculate the length of side (A) right triangle. This radius is twice that used in the previous formula, so triple it and also divide by the square root of triples: A = R*3/√3.
The perimeter (P) of an equilateral triangle to calculate the length of its side (A) is even simpler, because the lengths of the sides in this figure are the same. Just divide the perimeter into three pieces: A = R/3.
In an isosceles triangle the calculation of the length of the sides of known perimeter is a bit more complicated - you need to know more and a length of at least one of the parties. If you know the length of side A lying in the base of the figure, the length of any side (In) find halving the difference between the perimeter (P) and the size of the base: B = (R-A)/2. And if you know the side, the length of the base define by subtracting from the perimeter to twice the length of the side: A = R-2*V.
Knowledge area (S) occupied on the plane right triangle, is also sufficient for finding the lengths of its sides (A). Extract the square root of the area ratio and the square root of three, and the result double A = 2*√(S/√3).
In a right triangle, unlike any other, to calculate the length of one of the parties is sufficient to know the lengths of the other two. If the target side is the hypotenuse (C), to do this, find the square root of the sum of the lengths of the known sides (A and b) squared: C = √(A2+B2). But if you want to calculate the length of one of the other two sides, then the square root must be obtained from the difference of the squares of the lengths of the hypotenuse and another side: A = √(C2-B2).