The bisector of a triangle has a number of properties. If it is correct to use them, it is possible to solve problems of different level of complexity. But even having data on all three bisectors, it is impossible to construct a triangle.
What is a bisector
Studying properties of triangles and the solution of tasks, related – interesting process. It allows to develop at the same time both logic, and spatial thinking. One of important components of a triangle is the bisector. The bisector is a piece which leaves a corner of a triangle and divides it into equal parts.
In many tasks in geometry in conditions there are data on a bisector, at the same time it is required to find value of a corner or length of the opposite side and so on. It is necessary to find parameters of the bisector in other tasks. To define the correct answer of any of the tasks connected with a bisector it is necessary to know its properties.
Properties of a bisector
First, the bisector is a locus which are removed on equal distances from the parties adjacent to a corner. Secondly, the bisector of a triangle divides the party opposite to a corner into pieces which will be proportional to the adjacent parties. For example, there is ABS triangle, in it the corner of B leaves a bisector which connects vertex of angle to the M point on the adjacent party the EXPERT. After carrying out the analysis, we will receive a formula: AM/MS=AB/BS. Thirdly, the point which is crossing of bisectors from all corners of a triangle acts as the center of the circle entered in this triangle. Fourthly, if two bisectors of one triangle are equal, so this triangle is isosceles. Fifthly, if there are data on all three bisectors, then it is impossible to execute creation of a triangle even if to use compasses. Quite often for the solution of a task the bisector is unknown, it is necessary to find its length. To solve a problem, it is necessary to know a corner which it leaves and also lengths of the parties adjacent to it. In that case length of a bisector equals to the doubled work of the adjacent parties on a cosine of the angle, divided in half into the sum of lengths of the adjacent parties.
Rectangular triangle
In a rectangular triangle the bisector has the same properties, as in usual. But the additional property is added – the bisector of a right angle forms when crossing a corner in 45 degrees. Moreover, in an isosceles rectangular triangle the bisector which is lowered on the basis will also act as height and a median.