Bunuel wrote:
alphabeta1234 wrote:
Hey Bunuel,
When you extend a line from the vertex of a right angle to the hypotunse, each of the new two triangles are similar to the larger one. But are the two smaller triangles similar to one another?
Yes, the perpendicular to the hypotenuse will always divide the triangle into
two triangles with the same properties as the original triangle.
Generally if two triangles are similar to the third one, then they are naturally similar to each other.
Hey Bunuel,
Thank you very much for your response. I guess then as corrolary to the first question. Since all the angles are the same in all three triangles (let say angles x*,y*, and z*, where z*=90), Is there an easier way to identify which angle is referring to which side in the triangle within the equation you stated, AB/AC=AD/AB=BD/BC? Or are you mentally rotating the triangles making sure angle y* is opposite to AB, angle x* is opposite to AC, and checking it with angle y* opposite to AD, and angle x* is opposite to AB. Are you mentally rotating and checking that the all the sides are in conjunction with (opposite to x*)/(opposite to y*). Or is there an easier, more systematic method your using to ensure that all those sides are in fact in the same proportion.
On another side note, unrelated to my question above:
I also noticed that in your equation that each ratio is of sides of the same triangle equaling the ratio of the other triangle, opposite degree sides. So (Side 1, Triangle A)/(Side 2, Triangle A)=(Side 1 Triangle B)/(Side 2, Triangle B). I always imagined that if Triangle A (sides X,Y,Z) and Triangle B(sides M N P), where triangle A is bigger than B, then their similarity defined the following relationship:
X=Mc , where c is a some positive constant>1 and sides X and M are opposite to the same angle
Y=Nc
Z=Pc
Then the relationship would be defined as c=X/M=Y/N=Z/P= (side 1 Triangle A)/(side 1 Triangle B)=(side 2 Triangle A)/(side 2 Triangle B)=(side 3 Triangle A)/(side 3 Triangle B)
I just noticed that this relationship can always be manipulated to get c=X/M=Y/N ==> new constant=M*c/Y=X/Y=M/N=(Side 1 Triangle A)/(Side 2 Triangle A)=(Side 1 Triangle B)/(Side 2 Triangle B), which i think is very interesting. Same sides of the triangle are also in proportion to the same sides of the other triangle and another new constant the related the two triangles is embedded within the relationship of the triangles own sides!!! (eek, hope that made sense)
I also learned from you that if you had two similar triangles then (Area Triangle A)/(Area Triangle B)=(Side 1 Triangle A)^2/(Side 2 Triangle B)^2.
Are there any other key relationships that are important about similar triangles?