Question 571203
the altitude to the hypotenuse of a right triangle divides the right triangle into 3 similar triangles.
assume the altitude intersects the hypotenuse of triangle ABC at point D.
the 3 right triangles that are similar are:
ABC, ADB, CDB
since the triangles are similar, their corresponding angles are congruent and their corresponding sides are proportional.
if we label side AD equal to x and we label side CD equal to y, then we get a ratio of:
x/12 = 12/y
if we cross multiply, then we get:
xy = 12*12 which becomes:
xy = 144
since x + y = 26, we can solve for y to get:
y = 26 - x
we can substitute for y in the equation of:
xy = 144 to get:
x(26-x) = 144
simplify to get:
26x - x^2 = 144
add x^2 to both sides of this equation and subtract 26x from both sides of this equation to get:
x^2 - 26x + 144 = 0
this is a quadratic equation that factors to:
(x-8) * (x-18) = 0
this results in:
x = 8 or x = 18
if x = 8, then y = 18
if x = 18, then y = 8
this is because y = 26 - x.
we will allow x to be equal to 8 which results in y being equal to 18.
those are the 2 segments of the hypotenuse formed by the altitude.
the following diagram should help display what is happening.
<img src = "http://theo.x10hosting.com/2012/feb111.jpg" alt = "$$$" />
this is what is called the mean proportional in a right triangle.
a tutorial that discusses that can be found here:
<a href = "http://www.regentsprep.org/Regents/math/geometry/GP12/LMeanP.htm" target = "_blank">http://www.regentsprep.org/Regents/math/geometry/GP12/LMeanP.htm</a>