Median of a triangle is called the piece connecting any of triangle tops to the middle of the opposite side. Therefore the problem of creation of a median by means of compasses and a ruler comes down to a problem of finding of the midpoint.
It is required to you
- - compasses
- - ruler
- - pencil
Instruction
1. Construct a triangle of ABC. Let it is necessary to carry out a median from top With to the party of AB.
2. Let's find the middle of the party of AB. Establish a compasses needle in A point. Put other end of compasses in B point. Thereby compasses legs you measured AB length. Carry out a circle with the center in a point A and radius R equal to AB.
3. Then, without changing distance between compasses legs, establish a compasses needle in B point. Carry out a circle with the center in a point In and the same radius of AB.
4. The circles which are carried out from points And yes In have to be crossed in two points. Call them, for example, M and T.
5. Connect a line of a point of M and T. Tochk in which the piece of MT will cross AV piece, and will be AV midpoint. Let's call this point E. Kstati's point, direct MT will not only halve AV piece, but also to be a perpendicular to it. So if you are faced by a task to construct a perpendicular to a piece, act according to the same scheme, as for finding of the midpoint.
6. So, as E - the middle of the party of AV, the piece of SE will be the required median of a triangle which is carried out from top With to the party of AV. Connect by means of a line of a point With and E.
7. If it is necessary to carry out also medians from tops of a triangle And yes In to the parties of VS and the EXPERT respectively, do the similar procedure. Remember that all three medians of a triangle have to be crossed in one point.
8. Away from the drawing describe the actions. Consistently note that you build. What lines, circles you carry out and by what letters you designate the points received on crossings.
9. In tasks on construction by compasses and a ruler something usually is required not only to construct, but also to prove that the used sequence of actions resulted in the necessary result. On construction the quadrangle of AMVT is a rhombus (AM=BM=AT=BT=AB). A rhombus - a special case of a parallelogram. Parallelogram diagonals are halved by a point of intersection (property of a parallelogram). That is, the point E received on crossing of diagonals of a rhombus of AB and MT gives the middle of AV. Since a point E - the middle of AV, SE - AVS triangle median (by definition). This completes the proof.