Geometry Unit 8, sec 8.6 & 8.7
Proportions and Similar Triangles & Dilations - Guided notes
I). Proportions and Similar Triangles: Sec 8.6
If a line parallel to one side of a triangle intersects the other 2 sides, then it divides the 2 sides proportionally, and the two triangles are similar.
Example: Triangle ABC with line segment DE parallel to side AC, with point D on AB and point E on BC. Since DE is parallel to AC, then (DB)/(DA) = (BE)/(EC) and (BA)/(BD)=(BC)/(BE)=(AC)/(DE)
If BD = 7, AD = 3, BE = 8, ED = 6, EC = x, and AC = y, solve for x and y.
(DB)/(DA) = (BE)/(EC)
7/3 = 8/x
7x = 24
x = 24/7
(BA)/(BD) = (BC)/(BE) = (AC)/(DE)
10/7 = (8 + 24/7)/(8) = y/6
10/7 = y/6
7y = 60
y = 60/7
B) Theorem 8.7: If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other 2 sides.
Example: Given diagram - Triangle ABC with ray CD from vertex C bisecting angle ACB
If ray CD bisects angle ACB, then (AC)/(AD) = (CB)/(DB)
If AC = 6, AD = 3, DB = x and CB = 8 then
6/3 = 8/x
6x = 24
x = 4
II) Dilations: Sec 8.7 C) Dilations: Changes the size of a figure, so it is NOT an isometry. It has center, C, and a scale factor, k. 1. Reduction- the scale factor is between 0 and 1 (this means that 0 is less than the scale factor which is less than 1) 2. Enlargement - the scale factor is k>1. *NOTE: To find scale factor, divide Image by Preimage (CP’)/ (CP) = k = scale factor Example: Given center point C and preimage Triangle PQR that maps to image Triangle P’Q'R’, CP = 6 and CP’ = 3, then the scale factor is 3/6 or 1/2. Given center point C and preimage Triangle STU that maps to image Triangle S’T'U’, CS' = 10 and CS = 5, then the scale factor is 10/5 = 2/1. Remember that scale factors have to be in a ratio! D) Dilation in coordinate geometry: (x,y) maps to (kx, ky) where k is the scale factor. Example: If you have a point P (3,2) and the scale factor is 6, then (x, y) maps to (kx, ky) so… P (3,2) maps to P' (3 x 6, 2 x 6) = (18,12).
