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In [tex]\triangle ABC, AB=4, BC=5, and AC=7[/tex]. Point X is in the interior of the triangle such that [tex]AX^2 + BX^2 + CX^2[/tex] is a minimum. What is X, and what is the value of this expression?
What do you mean by the 'centre' of the triangle?Tide said:Then test it! The method I proposed places that point at the center of the triangle. Compare the values of f(x, y) at both points.
P.S. I haven't tested whether the point I found is a minimum or a maximum!
It's at the centroid. A triangle does not have mass and so does not have a "center of gravity"!maverick6664 said:yeah, and notice it's the center of gravity of the triangle
HallsofIvy said:It's at the centroid. A triangle does not have mass and so does not have a "center of gravity"!
(I am waging a hopeless war against using physics terms in mathematics.)
The inequality problem in Triangle ABC refers to finding the values of x and the minimum value in the triangle based on its three sides, A, B, and C, and the corresponding angles.
Finding x and the minimum value in Triangle ABC is important because it helps determine the validity of the triangle, as well as its size and shape. It also allows for the calculation of other important properties, such as area and perimeter.
The formula for finding x in Triangle ABC is based on the law of cosines, which states that c² = a² + b² - 2ab cos(C). This can be rearranged to solve for x: x = √(a² + b² - 2ab cos(C)).
The minimum value in Triangle ABC can be determined by finding the smallest side of the triangle. This can be done by comparing the lengths of the three sides and choosing the shortest one.
Yes, there are many real-life applications for solving the inequality problem in Triangle ABC. For example, it can be used in engineering and architecture to determine the stability and strength of structures, or in surveying to measure land and calculate distances.