If you have 3 numbers p, q, and r in such a way that
q^2 = pr,
then q is said to be the geometric mean between p and r
e.g. 4, 12, and 48
since 12^2 = 4*48 , 12 is the geometric mean between 4 and 48
(this is true for any 3 consecutive numbers in a geometric sequence)
Label your original right-angled triangle ABC, so that C = 90°
draw a perpendicular from C to AB, meeting it at D
(You admit to understanding that the triangles ACD and CBD are similar).
Label the angles x and o so you see which sides correspond with the
equal angles.
So corresponding sides are in the same ratios. ....
IF AD, DC , etc represents the lengths of the line segments, then
AD/DC = DC/BD
(DC)^2 = AD * BD
such showing that DC is the geometric mean for AD and BD
For the 2nd corollary use the original triangle ABC, which is also similar to each of the smaller ones, to set up ratios in a similar way
Can anyone explain the Corollaries to Theorem 7-3. (from the Pearson Geometry textbook) I understand the Theorem but not the Corollaries.
The Theorem states "The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other."
Corollary 1 states "The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse."
Corollary 2 states "The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to the leg."
If you can explain them to me that'd be great help. I'm not looking for any test answers by the way, because I want to understand.
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Yeah this took a bit for me to understand (but hey math takes some time lol). I think I have it down enough for me to continue studying. Anyway, tysm mathhelper!!
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