2. 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 Let's find the distance between two points. So the distance from (-6,4) to (1,4) is 7. (1,4)(-6,4) 7 units apart

3. 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 What coordinate will be the same if the points are located vertically from each other? So the distance from (-6,4) to (-6,-3) is 7. (-6,-3)(-6,4) 7 units apart

4. 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 But what are we going to do if the points are not located either horizontally or vertically to find the distance between them? Let's add some lines and make a right triangle. This triangle measures 4 units by 3 units on the sides. If we find the hypotenuse, we'll have the distance from (0,0) to (4,3) Let's start by finding the distance from (0,0) to (4,3) ? 4 3 The Pythagorean Theorem will help us find the hypotenuse 5 So the distance between (0,0) and (4,3) is 5 units.

5. 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 Now let's generalize this method to come up with a formula so we don't have to make a graph and triangle every time. Let's add some lines and make a right triangle. Solving for d gives us: Let's start by finding the distance from (x 1,y 1 ) to (x2,y2)(x2,y2) d x 2 - x 1 y 2 – y 1 Again the Pythagorean Theorem will help us find the hypotenuse (x 2,y 2 ) (x1,y1)(x1,y1) This is called the distance formula

6. Let's use it to find the distance between (3, -5) and (-1,4) (x1,y1)(x1,y1)(x2,y2)(x2,y2) 3 -5 4 found with a calculator Plug these values in the distance formula

7. Example 2 Determine if triangle ABC with vertices A(-3,4), B(5,2) and C(-1,-5) is an isosceles triangle. (Hint: An isosceles triangle must have at least 2 sides of equal length.) A(-3,4) B(5,2) C(-1,-5)

8. A(-3,4) B(5,2) C(-1,-5) BC and AC have the same length so triangle ABC is Isosceles.

9. 123 4 1 2 3 4 -2 -3 -4 5 -5 Zayed althani school Math department Mohamad badawi : hamadaa_math@yahoo.com Find the area of the circle in the figure أوجد مساحة الدائرة في الشكل Worksheet ( 1 ) The Distance Formula Hint : find r using the distance formula Remember that :

10. 123 4 1 2 3 4 -2 -3 -4 5 -5 Zayed althani school Math department Mohamad badawi : hamadaa_math@yahoo.com Draw and shade triangle ABC where : A ( -1, 3 ), B ( -2, -1 ), C ( 3, 2 ) then prove that the triangle is isosceles. ارسم وظلل المثلث الذي رؤوسه ثم برهن انه متساوي الساقين Worksheet ( 2 ) A ( -1, 3 ), B ( -2, -1 ), C ( 3, 2 ) The Distance Formula