To find the sides of the rectangle need to consider one of those right triangles. In it, the hypotenuse is the diagonal ofth rectangle, and the sides of his parties. Before the calculation of numeric values we need to find equations in General form. For each hand will be your equation. So, to get formulas in a right triangle, label each of the Latin letters a and b and the hypotenuse is C.
The solution of the problem consists of determining the sine and the Pythagorean theorem. Select any of the acute angles in the triangle (they are equal), with whom you will work. Define adjacent to it side and, opposite from it, the other leg. For example, suppose that adjacent to the angle, is side b, and opposite - side.
Further, based on the definition of the sine, which States that the sine of an angle in a right triangle is equal to the ratio of the opposite leg to the hypotenuse, write down the equation: sin 45 = a/C. In this example, the condition is known: the sine of the angle (sin 45 ~0,7) and hypotenuse s. Hence, we get the equation 0,7=a/C from where a=0,7 p. it Remains to substitute the numerical value of p. was Found side a will be equal to the parallel side of the rectangle. Thus, the two known hand shapes.
Advice 2: How to find the sides of the rectangle
A special case of a parallelogram - rectangle – only known in Euclidean geometry. Have a rectangle all the angles are equal, and each of them individually is 90 degrees. On the basis of private properties of the rectangleand from the properties of the parallelogram of the parallel opposite sides can find a hand figure by the given diagonals and angle from their intersection. Calculating sides of a rectangle based on additional constructions and apply the properties of the resultant shapes.
Draw a rectangle EFGH. Record the known data: the diagonal of the rectangle EG, and the angle α, obtained from the intersection of the two equal diagonals FH and EG. Build the figure and mark the diagonal between them the angle α.
The letter And mark the point of intersection of the diagonals. Consider the constructions formed by the triangle EFА. According to the property of the rectangle, its diagonals are equal and are bisected by the intersection point A. Calculate the values of FA and EA. Since the triangle is isosceles EFА and his side of EA and FA are equal and accordingly equal to half the diagonal EG.
Next, calculate the first side EF of the rectangle. This is the third unknown side of the considered triangle EFА. According to the theorem of the cosines of the appropriate formula, find the side EF. To do this, substitute in the formula for cosines of the previously obtained values of FA equal to the sides EA and the cosine of the known angle α between them. Calculate and record the obtained value of EF.
Find the second side of the rectangle FG. For this we consider another triangle EFG. It is rectangular, where the hypotenuse BC and the side EF. According to the Pythagorean theorem, find the second leg FG with the applicable formula.
In accordance with the properties of the rectangle, its opposite edges are equal. Thus, the side GH is equal to the side EF, and = FG. Write in the answer all the computed sides of the rectangle.