An arc is any connected part of a circle.The diameter of a circle is twice its radius. The diameter of a circle is length of a chord containing the center of the circle and is denoted by d.A chord of a circle is a line segment with endpoints on the circle.The circle with center P and radius r is the set of all points in the plane with distance r from P. Given a point P in the plane and let r be a positive real number.Converse of the Pythagorean Theorem : i f the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.Pythagorean Theorem : The square of the hypotenuse is equal to the sum of the squares of the legs in any right triangle.Two perpendicular lines have slopes that are negative recipro cals. We need to show that diagonals and are perpendicular using the equations of the lines L 1 and L 2 containing them. Since a rhombus is a parallelogram, we can use exactly the same set-up that we used in the previous proof. How do we prove that the diagonals of a rhombus are perpendicular? Similarly, if we apply it to B=(r, 0) and D=(s, t), we get M BD = P. If we apply the midpoint formula to A= (0,0) and C= ( r+s, t ), we get M AC= P. We can substitute x in either of the equations in order to find y. Because L 2 contains B = ( r, 0), the equation of L 2 is. Similarly, the slope m 2 of the line L 2 containing BD is. Recall that ( y – y 1 ) = m ( x – x 1 ) is the equation of the line with slope m containing the point ( x 1, y 1 ). Because L 1 contains the origin, the equation of L 2 is. The slope m 1 of the line L 1 containing is. We can find P by finding the equations of the lines containing and. If P denotes the point of their intersection, we want to show that P is the midpoint of both the segments and. is parallel to , so the points C and D will have the same y -coordinate, i.e., v=t. We can put any parallelogram ABCD in a plane, such that A is the origin and B is a point on the positive x -axis, i.e. The basic tools of coordinate geometry are the distance and midpoint formulas and equations of lines, all of which were discussed in previous lessons. How do we use coordinate geometry to prove that the diagonals of a parallelogram bisect each other?
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