Dropping an altitude from the right angle to the hypotenuse, we see that our desired height is (We can also check from the other side). Notice that we now have a 30-60-90 triangle, with the angle between sides and equal to. We have Solution 2īy the Pythagorean Theorem, we have that the length of the hypotenuse is. Dropping an altitude from the right angle to the hypotenuse, we can calculate the area in another way. By the Pythagorean Theorem, we have that the length of the hypotenuse is. We find that the area of the triangle is. How long is the third altitude of the triangle? So the red line goes from (0,0) to (3.84, 2.88).The two legs of a right triangle, which are altitudes, have lengths and. 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. Using the Altitude Theorem we in A B C with altitude A D we obtain: A D 2 B D C D and therefore B D 2 A D 4 C D 2 and C D 2 A D 4 B D 2. The hypotenuse line goes through the points (6,0) and (0,8). Power of a point (B) B M B D 2 A B and Power of a point (C) C N C D 2 A C. 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 Right Triangle Similarity Theorem If the altitude. In this triangle 6 is the hypotenuse and the red line is the opposite side from the angle we found. Have students share the problems that they developed in Activity 4.5.4. 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). Two important corollaries of Theorem 7-3 involve a geometric mean. 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, the altitude to the hypotenuse yields three similar triangles. 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. This altitude splits the hypotenuse into two segments of lengths d and e.
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