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GEOMETRIC MEAN

GEOMETRIC MEAN. Geometric Mean – the geometric mean between two numbers is the square root of their product. Ex: 1 Find the geometric mean between 4 and 9 x = √4∙√9 x = √36 x = 6. Ex 2: Find the geometric mean between 6 and 15. x = √6∙15 x = √90 x = √9∙10 x = 3√10. GEOMETRIC MEAN.

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GEOMETRIC MEAN

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  1. GEOMETRIC MEAN Geometric Mean – the geometric mean between two numbers is the square root of their product Ex: 1 Find the geometric mean between 4 and 9 x = √4∙√9 x = √36 x = 6 Ex 2: Find the geometric mean between 6 and 15. x = √6∙15 x = √90 x = √9∙10 x = 3√10

  2. GEOMETRIC MEAN Remember: Altitude – a segment extending from the vertex of a triangle to a the line containing the opposite side and perpendicular to that side. Similar Triangles – have the same shape, but not necessarily the same size. That is, corresponding angles are congruent, corresponding sides are proportional.

  3. Now let us consider a right triangle with an altitude drawn from the vertex of the right angle. Let's explore the triangle to see what happens. In this figure, segment BD is an altitude ofDABC. We also notice that the altitude divides the original, larger triangle into two smaller triangles. But what else can we discover about the the two triangles?

  4. The two new triangles are DADB and DBDC. Look at the angles in the new triangles that correspond to the angles in the first one.

  5. Theorem7.1 – If an altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the two triangles formed are similar to the given triangle. In our example DABC ~ DADB ~ DBDC

  6. Now, since we know about proportional sides in similar triangles, we can discover even more about these triangles. BD is the geometric mean between AD and DC. Theorem 7.2 – The measure of an altitude drawn from the vertex of the right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.

  7. Find x in the triangle. x is the altitude, so it is the geometric mean between the two sides. x = √(2)(5) x =√10

  8. Can we find the lengths of one of the legs? Theorem 7.3 – If the altitude is drawn from the vertex of a right triangle to its hypotenuse, then the measure of a leg of the triangle is the geometric mean between the measures of the HYPOTENUSE and the segment of the hypotenuse ADJACENT to that leg.

  9. Find y and in the triangle. The measure of the hypotenuse is 2 + 5 or 7. The length of the segment adjacent to y is 2. y = √(2)(7) x =√14 The length of the segment adjacent to z is 5. z = √(5)(7) z = √35

  10. HW # 4, p 346, 13-32 all

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