Diameter of a Circle Circle is defined as the set of points that is at an equal distant from the centre of the circle. There are a number of terminologies involved in a Circle. Some of them are as follows: Centre: The predetermined point from which the surface of the circle is at an equidistant is called the centre of a circle. Radius: The constant distance from the centre to a point on the surface of the circle is called its radius . Circumference: The boundary of a circle is called its circumference. Chord: A line segment whose end points is present on the circumference of a circle is called a chord . Diameter: A chord crossing through the midpoint of a circle is called its diameter. Know More About Adding and Subtracting Polynomials Worksheet
Circle Formulas Diameter of a Circle: Diameter = 2 X Radius Radius of a Circle: Radius(R) = Diameter / 2 Area of a Circle: Area = pi X R2 Circumference of a Circle: Circumference = 2 X pi X R Circle Theorem Theorem 1: A perpendicular from the centre of a circle to a chord bisects the chord. Given : AB is a chord in a circle with centre O. OC ⊥ AB. To prove: The point C bisects the chord AB. Construction: Join OA and OB Proof: In triangles OAC and OBC, m∠OCA = m∠OCB = 90 (Given) Learn More Area of a Circle Worksheet
OA = OB (Radii) OC = OC (common side) ∠OAC = ∠OBC (RHS) CA = CB (corresponding sides) The point C bisects the chord AB. Hence the theorem is proved. Theorem 2: AB and CD are equal chords of a circle whose centre is O. OM ⊥ AB and ON ⊥ CD. Prove that m∠OMN = m∠ONM.
What are Perpendicular Lines Two intersecting lines will have four angles formed at the intersection points. If all the four angles are equal, then the two lines are said to be perpendicular to each other. We already know by linear postulate theorem that the two vertically opposite angles are equal. Hence if these two lines are perpendicular, then all four angles are 90 degrees Examples of perpendicular lines: In the graph paper, The X-axis and Y-axis are perpendicular. In an ellipse two axes, minor axis, and major axis are perpendicular. For a line segment, any shortest line from a point outside the circle is perpendicular. Tangent and normal to any curve are perpendicular lines. Slopes of two perpendicular lines: In coordinate Geometry, when two lines are perpendicular, the product of the slopes of the lines is -1. This property has a lot of applications in finding the equation of perpendicular lines, length of perpendicular segment from a point to a given line, etc.
For any curve in a graph with equation y = f(x), the slope of the tangent is defined as the rate of change of y with respect to x at that point. The normal to this curve at this point is perpendicular to the tangent line. Example: In a circle, with centre at the origin and radius 3, the equation will be of the form (x)²+(y)² = 3². Take any point say (0,3). To find the tangent, we have to find dy/dx. Differentiating, 2x+2y =0 Hence, the slope of the normal is perpendicular to x axis or parallel to y axis. Example for Perpendicular Lines from a Point to a Line Let AB be a line with coordinates (1,2) and (3,4). Measure the length of perpendicular line from (-1,1) to this line segment. We know that the perpendicular line from (-1,1) has a slope of -1/slope of AB. Equation of AB is (x-1)/(3-1) = (y-2)/(4-2) Or x-1 = y-2 Or y = x+1 Slope of AB passing through (1,2) and (3,4) is 4 - 2/3 -1 =1. Slope of perpendicular line to AB is -1. Read More About Area of Irregular Shapes Worksheet
Since the perpendicular line passes through (-1,1) equation of the perpendicular is y-1 = -1(x+1) or y =-x -1 +1 or y = -x. To get the foot of the perpendicular line on AB, we solve the two equations by substitution method. y = x+1 = -x This on simplification gives 2x = -1 or x = -1/2. Since y = -x , we have y = +1/2, So, foot of the altitude from the point (-1,1) is (-1/2,1/2). The length of the perpendicular segment is between (-1,1) and (-1/2,1/2) is √[ (-1/2+1)²+(1/2-1)²] = √(1/4+1/4) = √(1/2) = 1/1.414 = 0.707 approximately.