Math Notes PDF

# APPLICATIONS OF DISTANCE FORMULA **1. Prove that the triangle whose vertices are A(-1,-1) B(3,-4) and C(2,3) is an isosceles.** * A(-1,-1) * B(3,-4) * C(2,3) **AB** * d = √(x2 - x1)² + (y2 - y1)² * d = √(3 - (-1))² + (-4 - (-1))² * d = √(3 + 1)² + (-4 + 1)² * d = √4² + (-3)² * d = √16 + 9 * d...

# APPLICATIONS OF DISTANCE FORMULA **1. Prove that the triangle whose vertices are A(-1,-1) B(3,-4) and C(2,3) is an isosceles.** * A(-1,-1) * B(3,-4) * C(2,3) **AB** * d = √(x2 - x1)² + (y2 - y1)² * d = √(3 - (-1))² + (-4 - (-1))² * d = √(3 + 1)² + (-4 + 1)² * d = √4² + (-3)² * d = √16 + 9 * d = √25 * d = 5 units **BC** * d = √(x2 - x1)² + (y2 - y1)² * d = √(2 - 3)² + (3 - (-4))² * d = √(-1)² + (3 + 4)² * d = √1 + 49 * d = √ 50 * d = 5√2 units **AC** * d = √(x2 - x1)² + (y2 - y1)² * d = √(2 - (-1))² + (3 - (-1))² * d = √(2 + 1)² + (3 + 1)² * d = √3² + 4² * d = √9 + 16 * d = √25 * d = 5 units **Conclusion** * Since AB is equal to AC * Therefore AABC is isosceles. # Coordinate of Midpoints of a Segment **1. Given points A and B with coordinates (9,9) and (-5,-5), respectively. What is the coordinate of its midpoint?** * A (9,9) * B (-5,-5) * M = (x1 + x2)/2 , (y1 + y2)/2 * M = (9 + (-5))/2 , (9 + (-5))/2 * M = (4)/2 , (4)/2 * M = (2,2) # Slopes of Parallel and Perpendicular Lines **Which pair of lines are parallel or perpendicular based on their equations?** 1. **2x + y + 7 = 0** **2x + y - 5 = 0** * y = -2x - 7 * m = -2 * y = -2x + 5 * m = -2 **Parallel Lines** * Slope = rise/run * m = (y2-y1)/(x2-x1) 2. **y = 3x + 5** * m = 3 3. **x + y = 10** * y = -x + 10 * m = -1 4. **2x + 3y = 6** * 3y = -2x + 6 * y = (-2/3)x + 2 * m = -2/3 # Equations and Graphs of Circles **A volcano at (0,0) has started to show signs of possible eruption. People residing near the area, specifically those within the 6km radius danger zone and are asked to evacuate. If Family A is located at 4km east and 8km south, is the family within the danger zone?** * d = √(x2 - x1)² + (y2 - y1)² * d = √(4 - 0)² + (-3 - 0)² * d = √(4)² + (-3)² * d = √16 + 9 * d = √25 * d = 5 km **Family A is **within** the danger zone.** **(x - h)² + (y - k)² = r²** (x - h)² + (y - k)² = r² **Equation of the Circle in Standard Form** # Ex. Write the equation of the circle. 1. **(4, -9) and r = 6** * (x - h)² + (y - k)² = r² * (x - 4)² + (y - (-9))² = 6² * (x -4)² + (y + 9)² = 36 2. **(-6, 3) and r = √2** * (x - h)² + (y - k)² = r² * (x - (-6))² + (y - 3)² = (√2)² * (x + 6)² + (y - 3)² = 2 3. **C (2, -5) r = √8** * (x - h)² + (y - k)² = r² * (x - 2)² + (y - (-5))² = (√8)² * (x - 2)² + (y + 5)² = 8 4. **C(9, 1) r = √30** * (x - h)² + (y - k)² = r² * (x - 9)² + (y -1)² = √30 * (x - 9)² + (y - 1)² = 30 # A review for the 3rd Quarter Test 1. **A. Give what is asked** * **i. What is the coordinate of the school?** * (-5, 6) 2. **How far is the school from the mall?** * d = √(x2 - x1)² + (y2 - y1)² 3. **What is the coordinate of M?** * M = (x1 + x2)/2, (y1 + y2)/2 4. **B. Give** * **(x + 4)² + (y - 3)² = 9** * Equation of a circle * C (-4, 3) * r = 3