Metric relations in a right-angled triangle
Geometry
 
1. PROJECTIONS
If we have a point P and a straight line r, P' is the projection of point P on line r, at the foot of the perpendicular line drawn from P to r. Line A'B' is the projection of line AB when the end points of line A'B' are the projection of points A and B.
Change the length and position of line AB using the control points A and B.


1.- Work out the length of the projection of a line 5 units long when it is perpendicular to the straight line r and when it is parallel to line r. Does the length of the projection change depending on the distance from the straight line?

2. THE PERPENDICULAR HEIGHT THEOREM
In a right-angled triangle the height drawn from the hypotenuse is the geometric mean of the two parts that it divides the hypotenuse into. The perpendicular height AD from hypotenuse BC has been drawn in the right-angled triangle ABC in the window. We can see that the following equality is true for any right-angled triangle:

The reason why this equality is true is because height AD divides the triangle ABC into two similar triangles whose sides are therefore in the same ratio. They are similar as their angles are equal (they are both right-angled triangles and the sides of the two acute angles are perpendicular to each other).

2.- Move control point A and watch how the values for the lengths of lines AD, BD and DC change but BD/AD and AD/DC stay in the same ratio.

3.- Alter the length of the hypotenuse and move point A to get a different right-angled triangle.

4.- Draw a right-angled triangle in your exercise book whose sides are 10, 8 and 6 units long. Draw the perpendicular height from the hypotenuse and check that the perpendicular height theorem above is true. Compare the values you get with those given in the Descartes window.

3. THE THEOREM OF THE SIDES ADJACENT TO THE RIGHT ANGLE
In a right-angled triangle each of the other two sides (b and c) is the geometric mean of the projection of this side (b' and c') onto the hypotenuse (a) and the hypotenuse itself. As was the case above, the perpendicular height drawn from the hypotenuse divides the triangle into two similar triangles and the following equalities are true:

Move control point A and watch how the values of the other two sides and their projections onto the hypotenuse change. Check that the values indicated above are always in the same ratio. Change the length of the hypotenuse and move point A to get a different right-angled triangle.

5.- The perpendicular height from the hypotenuse in a right-angled triangle divides the hypotenuse into two segments which measure 7 and 4. Apply the theorem of the sides adjacent to the right angle to work out the length of the other two sides.

6.- Check your results in the window. 

7.- Work out the area of this triangle.
       
           
  Miguel García Reyes
 
Spanish Ministry of Education. Year 2001
 
 

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