![]() Along with the use of trigonometric relationships, the altitudes of a triangle can be used to determine many characteristics of triangles. Each of the altitudes of a triangle forms a right triangle, and the altitudes of a triangle all intersect at a point referred to as the orthocenter. The base of a triangle is determined relative to a vertex of the triangle the base is the side of the triangle opposite the chosen vertex. Since all triangles have 3 vertices, every triangle has 3 altitudes, as shown in the figure below: An altitude of the isosceles triangle is shown in the figure below: In other words, an altitude in a triangle is defined as the perpendicular distance from a base of a triangle to the vertex opposite the base. In a triangle however, the altitude must pass through one of its vertices, and the line segment connecting the vertex and the base must be perpendicular to the base. In other geometric figures, such as those shown above (except for the cone), the altitude can be formed at multiple points in the figure. Altitude in trianglesĪltitude in triangles is defined slightly differently than altitude in other geometric figures. Note that the altitude can be depicted at multiple points within the figures, not just the ones specifically shown. So the red line goes from (0,0) to (3.84, 2.88).The dotted red lines in the figures above represent their altitudes. 4/3x + 8 = 3/4x (multiply both sides by 12 to eliminate fractions) To find the point of intersection, we can set the two equations equal to each other: Since its y-intercept is 0, the equation of the red line is: Since the red line is perpendicular to the hypotenuse line, its gradient must be 3/4 (negative reciprocal or m*m = -1). Therefore the equation of the line of the hypotenuse is: Therefore the gradient (slope) is -8/6 = -4/3. The hypotenuse line goes through the points (6,0) and (0,8). So we can set up a proportion with the long legs and the hypotenuses. ![]() We know they are similar because they both have an angle of 90 degrees and they share the angle at the point (6,0). I will work with the bottom triangle and the big triangle. The red line divides the big triangle into two smaller triangles, both of which are similar to the big triangle. *note, you could also do this finding the top angle and using the upper triangle formed by the red line In this triangle 6 is the hypotenuse and the red line is the opposite side from the angle we found. We see that this angle is also in a smaller right triangle formed by the red line segment. We know that the legs of the right triangle are 6 and 8, so we can use inverse tan to find the base angle. Pythagoras tells us that the hypotenuse is 10 (6^2 + 8^2 = 10^2), and we already know the area of the triangle is 24, so 24 = 0.5(10)(red line) –> 24 = 5x –> x = 4.8. So it is also possible to calculate the area by doing 0.5(hypotenuse)(red line). But the red line segment is also the height of the triangle, since it is perpendicular to the hypotenuse, which can also act as a base. In a right triangle, we can use the legs to calculate this, so 0.5(8)(6) = 24. Here they are given based on the hint question: There are many methods to finding the answer.
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