Introduction to Coordinate Proof

Name_________________________________________ Date____________________ Class___________________ LESSON 4-7 Reteach Introduction to Coordinate Proof A coordinate proof is a proof that uses coordinate geometry and algebra. In a coordinate proof, the first step is to position a figure in a plane. There are several ways you can do this to make your proof easier. Positioning a Figure in the Coordinate Plane Keep the figure in Quadrant I by using the origin as a vertex. Center the figure at the origin. Center a side of the figure at the origin. Use one or both axes as sides of the figure. Position each figure in the coordinate plane and give the coordinates of each vertex. 1. a square with side lengths of 6 units 2. a right triangle with leg lengths of 3 units and 4 units __________________________________________ 3. a triangle with a base of 8 units and a height of 2 units __________________________________________ 4. a rectangle with a length of 6 units and a width of 3 units __________________________________________ 4-54 __________________________________________ Name_________________________________________ Date____________________ Class___________________ LESSON 4-7 Reteach Introduction to Coordinate Proof continued You can prove that a statement about a figure is true without knowing the side lengths. To do this, assign variables as the coordinates of the vertices. a right triangle with leg lengths c and d Position each figure in the coordinate plane and give the coordinates of each vertex. 5. a right triangle with leg lengths s and t 6. a square with side lengths k __________________________________________ 7. a rectangle with leg lengths  and w __________________________________________ 8. a triangle with base b and height h __________________________________________ __________________________________________ 9. Describe how you could use the formulas for midpoint and slope to prove the following. Given: HJK, R is the midpoint of S is the midpoint of Prove: __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ 4-55 Practice B 1. 3. 2. Reteach 1. 3. Possible answer: ABCD is a rectangle with width AD and length DC. The area of ABCD is (AD)(DC) or (4)(6) = 24 square units. By the Midpoint Formula, the coordinates of E are Possible answer on graph above. = 2. (3, 0) and the coordinates of F are = (0, 2). The x-coordinate of E is the length of rectangle DEGF, and the y-coordinate of F is the width. So the area of DEGF is (3)(2) = 6 square units. Since 6 = (24), the area of rectangle Possible answer on graph above. DEGF is one-fourth the area of rectangle ABCD. 3. Practice C Possible answer on graph above. 1. 2. A40 Possible answer on graph above. 4. 9. Possible answer: Use the midpoint formula to find the coordinates of the midpoints R and S. Then use the coordinates and the formula for slope to find the slopes of and . Challenge Possi 1. Possible drawing: graph above. 5. 2. Possible answer: A parallelogram is formed. 3. Possible answer: A(2a, 2d), B(2b, 2e), C(2c, 0), D(0, 0) Possible answer on graph above. 4. Answers based on coordinates in Exercise 3: L(a + b, d + e), M(b + c, e), N(c, 0), P(a, d) 5. Answers based on coordinates in Exercise 4: , 6. Answers based on coordinates in Exercise 4: ; 6. 7. Possible answer on graph above. 7. and have the same slope, so they are ||. and also have the same length, so they are @. Quadrilateral LMNP is a parallelogram because if one pair of opposite sides of a quadrilateral is both || and @, then the quadrilateral is a parallelogram. Problem Solving 1. about 4.2 yd 2. about 338.9 m 3. Possible answer on graph above. 8. 4. Rachel had ridden farther, about 3.2 miles; Malik had ridden about 2.2 miles. 5. D 6. J Reading Strategies 1. with one side along one of the axes A41 Name_________________________________________ Date____________________ Class___________________ 2. with the side you want to find the midpoint of along an axis with the midpoint at the origin 4-55