Midpoint and Length Applications

Basic Facts that help! Slope is rise/run = (y2 – y1) / (x2 – x1) slope of a vertical line is ∞ (undefined/denominator = 0) Slope of a horizontal line is zero (numerator is zero) Parallel lines have the same slope Perpendicular lines have slopes that are negative reciprocals of each other (2 and – ½ ) y = 2x + 5 and y = 2x – 5 are ________________ y = 2x – 8 and y = - ½ x + 4 are ________________________

Triangles Equilateral triangles have all side lengths equal Isosceles triangles have two side lengths equal Scalene triangles have no side lengths equal Triangle Problem Solving If you know the coordinates of the vertices then: Use length formula to find side lengths to identify triangle type. Use slopes of sides to see if a right angle is present.

Quadrilaterals If you know the coordinates of the vertices of a quadrilateral; the lengths and slopes will allow you to identify the type: Parallelogram has both pairs of opposite sides parallel. Rectangle has same properties as a parallelogram and the adjacent sides are perpendicular. Square has the same properties of a rectangle with all sides of equal length. Rhombus has both pairs of opposite sides parallel and all sides equal.

Midpoint of a line 𝑀= 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2 The midpoint of a line segment is the average of the 2 x-values and the average of the 2 y-values, expressed as an ordered pair. (x1, y1) (x2, y2) 𝑀= 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2

Circle around (0, 0) A circle with its centre at the origin has a radius of r, so it’s equation can be found by finding the length of a line segment connecting the centre and the outside edge. 𝑟= ( 𝑥−0) 2 + (𝑦−0) 2 r2 = x2 + y2 is the general equation for all circles!! P (x, y)

Class/Home work Page 71 # 1-6 – Math Power 10

AG2 and AG3 AG2 - solve problems using analytic geometry involving properties of lines and line segments; AG3 - verify geometric properties of triangles and quadrilaterals, using analytic geometry.

Q1 AG2 Prove that the midpoint M = (4, 2) is actually the midpoint of AB where A = (1, -2) and B = (7, 6).

Q2 AG2 Write the equation of a circle with centre (0, 0) and a radius of 4 cm.

Q3 AG2 Given a line segment ST has S = (6, 2) and T = (0, y) has a midpoint M = (x, 0): find x and y.

Q4 AG3 Classify the triangle that has vertices L (-7, 0), M (2, 1), N (-3, 5)

Q5 AG3 A building needs to be 45 m x 30 m. Jr engineers have marked corners as P (6, 36), Q (- 30, 9), R ( -12, -15), S (24, 12). Prove it’s a rectangle with 45 m x 30 m dimensions.

Q6 AG3 Outline the steps to solve the following problem: Determine the lengths of the line segments connecting the midpoints of the sides of a triangle with vertices (1, -1), (5, -8) and (3, 7).

Q7 AG3 Verify that the diagonals of a quadrilateral with vertices: (0, 0), (0, 8), (8, 0) and (8, 8) bisect each other perpendicularly.

Text work Math Power 10 Page 71 #7 - 15 page 95 #1 – 5, 9 – 13, 21, 28 Principles of Math 10 Pg 78 #10, 12 Pg 89 #8 Pg 124#2, 3, 4, 6, 7, 10, 13, 18* pg 142 #2, 4, 5, 10, 11