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Video Notes 7.4/7.5: Similarity & Proportionality in Right Triangles for Geometry

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alright hello everybody this is your teacher Miss Indiana and piece of your baby notes for section 7.475 triangles and proportionality and triangles again sorry about that you can fix that we're Smith said if you'd like to jump right into it we have the first 7.3 it is a no-name here is the diagram and they are three triangles that are all similar by simply dropping an altitude of a triangle is a right so you can see that here in angle be the right triangle and an altitude is dropped to the hypotenuse who has we dropped from the vertex where the right angle is at then you can create three similar triangles triangles vmc and triangle a b c r o similar text go to the next room or the next concept next concept is called a geometric mean we'll talk more about that in class just give me the definition so for any two numbers X and Y doesn't to be a coordinate pair and I wrote it here looking like a ordered pair of ceasing mean is the square root of their slim to metric mean of two numbers X and Y is equal to the square root of x times y but by the way in finance Wall Street will use the geometric mean to calculate things like the gross interest from their Investments stuff like that but we are going to take a look at how this applies into the world of geometry of course is really a set of corollaries Matthew forgot what a corollary is number 107.3 in other words imagine a spin-off from a different theorem so we have theorem 7.3 which already taught you and now we have this other idea that comes from that theorem anyways number 107.3 you can see the diagram here it's still a right triangle and we're dropping an altitude to the hypotenuse we're taking a look at these segments that are being split by the altitude work on that X work on that why and that altitude were labeling it a as you can see that altitude we can measure that altitude look at it's the geometric mean of X and Y so the length of an altitude to the hypotenuse is the geometric mean of the length of the segment of the hypotenuse you can see that here you know something about in terms of a proportion so this segment is to a as a is to why if you cross multiply you will get that are you close sort of XY next corollary to a theorem 7.3 I just a second one again identical diagram you have a right triangle there's a right angle you're dropping an altitude and you have two segments X and Y just what we had before we're not looking at actual altitude this time we're looking at the legs of this original right triangle with some calling L1 and L2 and we're also calculated parts here's the words you're the worst leg a triangle are the geometric means to the hypotenuse and the segment of the hypotenuse adjacent to that leg for example dislike L1 look at that is √ there x + y * x what is a tax is adjacent to L1 if I wanted to calculate L2 l x y if you can see here take a look at the next concept this is really section 7.5 now and we have the side splitter which you seen before looks like that the line is parallel to one side of a triangle so this line imagine this line right here with triangle and yellow line in the middle and it's going to intersect two of the sides of the triangle is parallel to the one of the side of the triangle has sides or split proportionally in other words C-sections XY and be so you can say that X is y as 8 is to be 60 wide X is to y as a is to be on the bottom to the top or bottom this theorem every so often you have three lines three lines are parallel are proportional eggs and I can you have three lines parallel and you have two transversals and you have enough all this X y a b r i didn't even say this segment the second is equal to this segment after these two seconds right around with that all right take a look at the last concert here triangle angle bisector 7.5 of a triangle can you have an angle bisector Meredith bisects the angle this these two angles are congruent now we created certain sections we have a b here we are BC over there we have leg ad and we have silent DC this is this is this Section 8 ABS to Ada BC is to the see the words sound like this if a ray bisects an angle of a triangle then it divides the opposite side into two segments that are proportional to the size of triangle in triangle angle bisector

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Video Notes 7.4/7.5: Similarity & Proportionality in Right Triangles for Geometry

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