Squaring, adding, simplifying, then taking square root gives Example 4 Note that both and have a common factor of. The distance from to is, by the distance formula: Recall that the coordinates of the foot of the altitude from vertex are given by: PROVE that the length of the altitude from vertex is: This can also be checked by the regular, preferred method (that is, finding the equation of side and the equation of the altitude from, then finding their point of intersection). So it’s verified that the altitude’s foot is at. Next, the “common denominator” in the simplified expressions for and is. įrom the diagram above, the foot of the perpendicular from vertex appears to be the point.
Find the coordinates of the foot of the perpendicular from vertex. Similarly, by dividing every term in the expression for by, we obtain the required simplified form.įrom these simplified expressions, we see that once we find the slope of the side where the “foot” of the altitude lies, then the rest is easy.
(Note that, in view of the original diagram,, and so division by doesn’t result in any absurdity.) To see why the first one is true, simply divide every term in the equationīy. The above equivalent representations are more convenient for computations. PROVE that the coordinates of the foot of the altitude in the above diagram can be re-written as. This way we have:ĭue to the conspicuousness of the terms and in the above expressions, both and can be recast into a different form. The coordinates of can be obtained by solving the equations of and simultaneously. Do you understand the title? The foot of an altitude
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