**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).**