Sam+M

Sam M Second Quarter Project Hour 3

This is the acute triangle that I created with the coordinates (1,2), (-3,-2), and (2,-5).

To find the midpoints D and E, I averaged the X values and the Y values.

Work and Answers: This is the acute triangle with the midpoints D and E.
 * D (midpoint of line segment AC):**
 * X coordinate**: (1+2) / 2 = **3/2 (1 1/2)**
 * Y coordinate**: [2+(-5)] / 2= **-3/2** **(-1 1/2)**
 * E (midpoint of line segment BC):**
 * X coordinate:** (-3+2) / 2 = **-1/2**

__**Slopes**__
 * of line segment AC:**
 * coordinates**: (1,2) (2, -5)
 * work**: [2-(-5)] / (1-2) = **7/-1 = -7**
 * Negative Reciprocal= 1/7**

__**Perpendicular Bisectors**__ y = 1/7x + b -3/2 = 1/7(3/2) + b -3/2 = 3/14 + b -3/14 -3/14 y = 5/3x + b -7/2 = 5/3(-1/2) + b -7/2 = -5/6 + b +5/6 +5/6 -8/3 = **-2 2/3 = b** This is the acute triangle with two perpendicular bisectors that bisect side AC and side BC.
 * of line segment BC:**
 * coordinates:** (-3, -2) (2, -5)
 * work:** [-2-(-5)] / (-3-2) = **3/-5 = -3/5**
 * Negative Reciprocal= 5/3**
 * of line segment AC:**
 * equation with negative reciprocal:**
 * midpoint D coordinates (3/2, -3/2) substituted:**
 * -12/7 = b**
 * of line segment BC:**
 * equation with negative reciprocal:**
 * midpoint E coordinates (-1/2, -7/2) substituted:**

__**Solving System of Equations**__ y = 5/3x -2 2/3 y = 1/7x -12/7 5/3x-2 2/3 = 1/7x-12/7 -1/7x -1/7x 32/21x-2 2/3 = -12/7 +2 2/3 +2 2/3 __32/21x__ = __20/21__ 32/21 32/21 y = 1/7(5/8) -12/7 y = 5/56 - 12/7 This is the acute triangle with the circumcenter now found and labeled as point F (5/8, -13/8).
 * Work:**
 * x = 5/8**
 * y = -13/8**
 * coordinates of F: (5/8, -13/8)**

__**Finding Distance from Vertex to Circumcenter**__ (5/8, -13/8) - (1,2) = (-3/8, -29/8) (-3/8)(-3/8) = 9/64 (-29/8)(-29/8) = 841/64 9/64 + 841/64 = 850/64 Square Root of 850/64 = **about 3.64**
 * Distance between F and A:**

(5/8, -13/8) - (-3, -2) = (29/8, 3/8) (29/8)(29/8) = 841/64 (3/8)(3/8) = 9/64 841/64 + 9/64 = 850/64 Square Root of 850/64 = **about 3.64**
 * Distance Between F and B:**

(5/8, -13/8) - (2, -5) = (-11/8, 27/8) (-11/8)(-11/8) = 121/64 (27/8)(27/8) = 729/64 121/8 + 729/8 = 850/64 Square Root of 850/64 = **about 3.64** This is the final picture, with the acute triangle circumscribed.
 * Distance Between F and C:**