Simplify (Make the Radicand As Small As Possible)

Radicals

Perfect Squares

Reducing Radicals Simplify (make the radicand as small as possible)

1) 2) 3) 4) 5)

Add and Subtract Radicals Simplify 1) 2) 3)

Multiply and Divide Radicals 1) 2)

Simplify a) b) Length of a Line Segment

How long is AB?

In general, how long is AB?

Side Note: |x| absolute value sign (means distance)

Example: Find the length of |PM|, where P(-2,1) and M(4,-3)

p. 71 #1a, 5a, 8, 10, 18, 19 Review from Grade 9 Equations of Lines

Example 1: Write an equation in standard form for the line that passes through the points C(8,2) and D(4,7)

Example 2: Write the equation of the line perpendicular to 3x + y - 6 = 0 and passing through P(5,2)

Practice p. 84 #1 - 3 p. 85 #1adgh, 2adgh p. 87 #1, 2 Equation of a Circle

An equation for the circle with centre at the origin and radius is

An equation for the circle with centre at (a,b) and radius r is

______

Example 1: Consider the circle defined by the equation

Complete the table of values to help visualize the graph.

Example 2: Determine an equation for each circle. (a) (b) (c)

Example 3: For each equation, state the radius of the corresponding circle and give the coordinates of one point on the circle. (a) (b)

Example 4: Find an equation for the circle that is centred at the origin and passes through (5, 2).

Example 5: Find an equation for the circle that is centred at (9,-7) and passes through (5, 2).

Example 6: Determine whether (2, -6) rests on the circle defined by , or is on the inside or the outside of the circle. Midpoint of a Line Segment

Definition: Midpoint

Example: Find the midpoint of line segment AB.

For any two points (x1, y1) and (x2, y2), the midpoint is ...

Use the formula to find the midpoint of AB again.

Example: A circle is centred at (5,4). One endpoint of a diameter drawn on the circle is (-1,2). What is the other endpoint of the diameter?

p. 77 #1aegh, 2, 3, 4, 7, 8, 10, 13 19ad, 20 Geometric Figures

Formulae: A(x1, y1) B(x2, y2)

Length Midpoint Slope

Triangles Quadrilaterals

Example: Consider A(-1,2), B(0,6), C(8,4), and D(7,0). What kind of quadrilateral is ABCD?

p. 95 #1-5 Geometric Figures Example 1) Verify that XYZ is a right triangle. 2) Classify the triangle as scalene, isosceles or equilateral.

p. 96 #12, 13, 20, 25 - 28, 30 Geometric Figures Continued

Right Bisector

Example: Find the equation of the right bisector to AB A(5,4) B(-1,2)

Median

Example: Find the equation of the median from A in triangle ABC. A(4,1), B(-1,-5), and C(5,-7)

Altitude

Example: Find the equation of the altitude from A in triangle ABC. A(4,1), B(-1,-5), and C(5,-7)

Practice p. 96 #6, 14, 22 Triangle Centres

geogebra.org

Centroid –

Circumcentre –

Orthocentre –

Consider triangle A(0,0), B(6,4) , C(12,-4).

Example 1: Verify that the centroid is (6,0)

Example 2: Determine the coordinates of the orthocentre.

p. 96 #15, 16, 18, 21 Distance from a Point to a Line

Along what line should you walk to reach the road in the shortest distance possible?

How far are you from the road if you are at P(-1,3) and the road is x + y - 5 = 0?

1) slope of the road 2) slope of walking line

3) equation of walking line 4) The point where you hit the road

5) Distance from you to where you hit the road

Practice: p103 #1a, 3ae, 4, 8, 9