![]() ![]() Ī division by two finally yields the formula of the geometric mean theorem. | C D | | D E | = | A D | | D B | ⇔ h 2 = p q. Geometric mean theorem as a special case of the chord theorem: Since the altitude is always smaller or equal to the radius, this yields the inequality. Now the altitude represents the geometric mean and the radius the arithmetic mean of the two numbers. For the numbers p and q one constructs a half circle with diameter p+q. Īnother application of provides a geometrical proof of the AM–GM inequality in the case of two numbers. The method also allows for the construction of square roots (see constructible number), since starting with a rectangle that has a width of 1 the constructed square will have a side length that equals the square root of the rectangle's length. Due to Thales' theorem C and the diameter form a right triangle with the line segment DC as its altitude, hence DC is the side of a square with the area of the rectangle. Then we erect a perpendicular line to the diameter in D that intersects the half circle in C. Now we extend the segment q to its left by p (using arc AE centered on D) and draw a half circle with endpoints A and B with the new segment p+q as its diameter. For such a rectangle with sides p and q we denote its top left vertex with D. So it's okay to have an altitude that is not inside your triangle.The latter version yields a method to square a rectangle with ruler and compass, that is to construct a square of equal area to a given rectangle. If I look at the other two altitudes in this obtuse triangles, we're going to have one altitude going like that I'm going to have to extend that side as well and we'll drop down another altitude. ![]() Notice that I had to extend that opposite side. So if we pick this vertex, our opposite sides are over here but that opposite side doesn't continue to where this altitude will drop. So a third case is the obtuse triangle, and here is where I say to a line containing the opposite side. However if I pick my 90 degree angle as my vertex, then we'll be able to see that altitude inside the triangle. If I pick this vertex right here the altitude will just be that leg of the triangle. That's going to be that leg of the triangle. If we look at a right triangle over here we can see that if I pick this vertex right here, we already have an altitude drawn. Notice that all three altitudes are inside the triangle. We would have two more altitudes, each of which would go perpendicular to the opposite side. So if I were to pick this top vertex right here, the altitude would go straight down perpendicular to the opposite side. So if we look at an acute triangle, there are going to be three altitudes, one form each vertex. It's not always to the opposite side and you're going to see why in a second here. So this definition is written very carefully. What we're talking about is a perpendicular segment, remember this symbol right here means perpendicular-I'm trying to get you used to seeing these symbols-from a vertex to a line containing the opposite side. When we're talking about triangles, there's a special segment three in each triangle called an "Altitude." So we're not talking about skiing here. ![]()
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