How to find tops of corners

How to find tops of corners

Proceeding from one point, straight lines form a corner where the general for them the point is top. Tasks when it is necessary to find coordinates of this top quite often occur in the section of theoretical algebra that then to define the equation of the straight line passing through top.


1. Before beginning process of finding of coordinates of top, decide on basic data. Accept that the required top belongs to a triangle of ABC in which coordinates of two other tops and also numerical values of corners, equal "e" and "k" are known for the party of AB.

2. Combine the new system of coordinates with one of the parties of a triangle of AB so that the beginning of a system of coordinates coincided with a point of A which coordinates are known to you. The second top of B will lie on OX axis, and its coordinates are also known to you. Determine by an axis OH value of length of the party of AB according to coordinates and accept it equal "m".

3. Lower a perpendicular from unknown top of C on an axis OH and on the party of a triangle of AB respectively. The turned-out height of "y" also defines value of one of coordinates of top of the C in axis OY. Accept that height of "y" divides the party of AB into two pieces equal "x" and "m – x".

4. As values of all corners of a triangle are known to you, so also values of their tangents are known. Accept values of tangents for the corners adjoining the party of a triangle AB equal to tan (e) and tan(k).

5. Enter the equations for two straight lines passing on the parties of AC and BC respectively: y = tan(e) * x and y = tan(k) * (m – x). Then find crossing of these straight lines, using the transformed equations of straight lines: tan(e) = y/x and tan(k) = y / (m – x).

6. If to accept that tan(e)/tan(k) equals (y/x) / (y/(m – x)) or after reduction of "y" - (m – x) / x, as a result you receive the required values of coordinates equal x = m / (tan(e)/tan(k) + e) and y = x * tan(e).

7. Substitute values of corners (e) and (k) and also the found value of the party of AB = to m in the equations x = m / (tan(e)/tan(k) + e) and y = x * tan(e).

8. Transform the new system of coordinates to the initial system of coordinates as between them one-to-one correspondence is established, and receive required coordinates of top of a triangle of ABC.

Author: «MirrorInfo» Dream Team