is there any short form of the given theorem
If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.
Answers
It is common practise to label the vertices of a triangle with capital letters and the sides with small letters.
It is also common practice to label the side opposite angle A with a small a, the side opposite angle B with a small b and the side opposite angle C with a small c ( see the diagram ).
The sides that form the arms of the angle A are said to be adjacent to A. The side on which the triangle stands is called the base of the triangle.
The sum of the angles of a triangle is 180° . This can easily be seen by drawing a straight line through the angle B and parallel to the side b. (see diagram).
The angles formed with this line are equal to A, B and C ( by the rule that alternate angles between parallel lines are equal).
Also A + B + C = 180° as together they make a straight line .
The straight line from angle B perpendicular to the base line b is called the height of the triangle. The height is labelled h in the diagram below.
You have previously learned that the area of a triangle is given by the formula.
Area F = ½×b×h
The letter G is used here to label the point where the height and the base intersect. This point is sometimes called the perpendicular projection of the point B onto the line b.Two triangles are said to be similar if all the angles of one triangle are equal to all the angles of the other. If we want to show that two triangles are similar it is sufficient to show that two angles are equal. If two angles are equal it is obvious that the third angle in each must be equal. The triangles in the above diagram are similar. It follows that the ratios,between corresponding sides are the same. We will now do some examples using these ratios.