The altitude can be inside the triangle , outside it, or even coincide with one of its sides, it depends on the type of triangle it is:. Where is the orthocenter located? The altitude h of the equilateral triangle or the height can be calculated from Pythagorean theorem. Applying the Pythagorean theorem :. And we obtain that the height h of equilateral triangle is:. Another procedure to calculate its height would be from trigonometric ratios.
The altitude h of the isosceles triangle or height can be calculated from Pythagorean theorem. By Pythagorean theorem :. In a isosceles triangle , the height corresponding to the base b is also the angle bisector , perpendicular bisector and median.
And now let's draw another line that is parallel to this line right over here, but it goes through this vertex. It goes through the vertex that's opposite that line. And so let me just draw it. And you can always construct these parallel lines just like that. And let's see what happens. So once again, these two lines are parallel. So you could view this green line as a transversal.
If this green line is a transversal, this corresponding angle is this angle right over here. If we view this green line as a transversal of both of these pink lines, then this angle corresponds to this angle right over here.
If we view this yellow line as a transversal of both of these pink lines-- actually, let's look at it this way. View the pink line as a transversal of these two yellow lines, then we know that this angle corresponds to this angle right over here. And if you view this yellow line as a transversal of these two pink lines, then this angle corresponds to this angle right over here.
And then the last thing we need to think about is if we think about the two green parallel lines and you view this yellow line as a transversal, then this corresponding angle in orange is right over here.
This corresponds to that angle, because this yellow line is a transversal on both of these green lines. So what I've just shown starting with this inner triangle right over here is that if I construct these parallel lines in this way, that I now have four triangles if I include the original one, and they're all going to be similar to each other.
And we know that they're all similar because they all have the exact same angles. You just need two angles to prove similarity. But all four of these triangles have the exact three angles. Now, the other thing we can show is that they're congruent. So all of these four are similar. And we also know they're congruent. For example, this side right over here in yellow is the side in this triangle, between the orange and the green side, is the side between the orange and the green side on this triangle right over here.
So these two-- we have an angle, a side, and an angle. Angle-side-angle congruency. So these two are going to be congruent to each other. Then over here, on this inner triangle, our original triangle, the side that's between the orange and the blue side is going to be congruent to the side between the orange and the blue side on that triangle.
Once again, we have angle-side-angle congruency. So this is congruent to this, which is congruent to that. All of these are going to be congruent. And by the same exact argument, this middle triangle is going to be congruent to this bottom triangle.
You have an angle, blue angle, purple side, green angle. Blue angle, purple side, green angle. They're congruent to each other. So you have all of these triangles are congruent to each other. So their corresponding sides are equal. So if you look at this triangle over here, we know that the side between the blue angle and the green angle is going to be equal to this angle right over here.
Sorry, equal to this length. So it's going to be equal to this length. Between the blue and the green we have this length, between the blue and the green we have that length, between the blue and the green we have that length right over there.
You may see "the altitude of the triangle is 3 centimeters". In this sense it is used in way similar to the "height" of the triangle. In most cases the altitude of the triangle is inside the triangle, like this: Angles B, C are both acute However, if one of the angles opposite the chosen vertex is obtuse , then it will lie outside the triangle, as below. Angle C is obtuse The altitude meets the extended base BC of the triangle at right angles.
In the animation at the top of the page, drag the point A to the extreme left or right to see this. It turns out that in any triangle, the three altitudes always intersect at a single point, which is called the orthocenter of the triangle. For more on this, see Orthocenter of a triangle. The following two pages demonstrate how to construct the altitude of a triangle with compass and straightedge.
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