Theorem 4-5
If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base
Theorem 4-5, also known as the Intermediate Value Theorem, states that if a continuous function f(x) takes on values f(a) and f(r) at two points a and r on an interval [a, r], then it must take on every value between f(a) and f(r) at least once on the interval [a, r]. In other words, if there is no point on the interval where the function jumps over a value between f(a) and f(r), then every value between f(a) and f(r) must be represented on the interval.
This theorem is useful in a variety of applications, such as finding roots of equations or proving the existence of solutions to certain problems. It relies on the key assumption that the function is continuous on the interval; without continuity, it may be possible for the function to jump over values and violate the intermediate value property.
To apply the Intermediate Value Theorem, one must first verify that the function is continuous on the interval [a, r]. Then, one must identify values f(a) and f(r) and show that there is no point on the interval where the function skips over values between f(a) and f(r). If these conditions are met, one can conclude that the function must take on every value between f(a) and f(r) at least once on the interval [a, r].
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