Altitude geometry activities12/21/2023 ![]() Many students told me that the hypotenuse is always √2 times longer than the leg "because the Pythagorean Theorem says so." After students used the Pythagorean Theorem to find the length of the diagonal of the square, I asked them to tell me the relationship between the legs and the hypotenuse of a 45-45-90 triangle. I told students that each side of the square was 1 cm and they had to use the Pythagorean Theorem to find the length of the diagonal. They had to use the Pythagorean Theorem to solve for the length of the diagonal (square) and the altitude (equilateral triangle). ![]() So, I had students discover the rules for 45-45-90 and 30-60-90 triangles by splitting a square in half and an equilateral triangle in half. ![]() Newell, why is the hypotenuse in a 45-45-90 triangle always √2 times longer than the legs? Why isn't it only 2 times the leg, since 45 is half of 90?" This question made me reflect on my teaching and it made me realize that I was not teaching the "why" in special right triangles. Theorem 64: If an altitude is drawn to the hypotenuse of a right triangle, then it is the geometric mean between the segments on the hypotenuse.Įxample 1: Use Figure 3 to write three proportions involving geometric means.įigure 3 Using geometric means to write three proportions.Įxample 2: Find the values for x and y in Figures 4 (a) through (d).įigure 4 Using geometric means to find unknown parts.īecause it represents a length, x cannot be negative, so x = 12.This is my third year teaching Geometry and every year, students have a hard time with special right triangles. This proportion can now be stated as a theorem. Theorem 63: If an altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and its touching segment on the hypotenuse. These two proportions can now be stated as a theorem. This produces three proportions involving geometric means. Note that AB and BC are legs of the original right triangle AC is the hypotenuse in the original right triangle BD is the altitude drawn to the hypotenuse AD is the segment on the hypotenuse touching leg AB and DC is the segment on the hypotenuse touching leg BC.īecause the triangles are similar to one another, ratios of all pairs of corresponding sides are equal. They have been drawn in such a way that corresponding parts are easily recognized.įigure 2 Three similar right triangles from Figure (not drawn to scale). Theorem 62: The altitude drawn to the hypotenuse of a right triangle creates two similar right triangles, each similar to the original right triangle and similar to each other.įigure 2 shows the three right triangles created in Figure . The following theorem can now be easily shown using the AA Similarity Postulate. In Figure 1, right triangle ABC has altitude BD drawn to the hypotenuse AC.įigure 1 An altitude drawn to the hypotenuse of a right triangle.
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