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.
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
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!!
Certainly! I'd be happy to explain the corollaries to Theorem 7-3.
Corollary 1 states that the length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments into which the hypotenuse is divided by the altitude. To understand this, we need to know what the geometric mean is.
The geometric mean of two numbers is the square root of their product. In this case, if we have a right triangle with a hypotenuse that is divided into two segments by an altitude, the length of the altitude can be found by taking the square root of the product of the lengths of the two segments.
Let's say the lengths of the two segments are 'a' and 'b', and the length of the altitude is 'x'. According to Corollary 1, we have the equation:
x = √(a * b)
This means that the length of the altitude is equal to the square root of the product of the lengths of the segments.
Corollary 2 states that the altitude to the hypotenuse of a right triangle separates the hypotenuse into two segments, such that 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.
To understand this, let's again consider a right triangle with an altitude to the hypotenuse. If we label the hypotenuse as 'c', and the lengths of the two segments into which it is divided by the altitude as 'a' and 'b', then Corollary 2 says that the lengths of the legs of the triangle, let's call them 'p' and 'q', can be found using the following equations:
p = √(a * c)
q = √(b * c)
In other words, the length of each leg of the triangle is equal to the square root of the product of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to the leg.
I hope this explanation helps you understand the corollaries to Theorem 7-3 better! If you have any further questions, feel free to ask.