Here we will prove that the tangent at any point of a circle and the radius through the point are perpendicular to each other.
A circle with centre O in which OP is a radius. XPY is a tangent drawn to the circle at the point P. The shortest line segment, drawn to a line from a point outside it, is perpendicular to the line.
Like Article. Last Updated : 05 May, Recommended Articles. Theorem - The lengths of tangents drawn from an external point to a circle are equal - Circles Class 10 Maths. Theorem - There is one and only one circle passing through three given non-collinear points Class 9 Maths. Radius of a circle having area equal to the sum of area of the circles having given radii. Check if two circles intersect such that the third circle passes through their points of intersections and centers.
Theorem - Angle opposite to equal sides of an isosceles triangle are equal Class 9 Maths. Article Contributed By :. Easy Normal Medium Hard Expert. A common internal tangent intersects the segment that joins the centers of the two circles. A common external tangent does not intersect the segment that joins the centers of the two circles.
How can we tell whether we are dealing with a common internal tangent, or common external tangent? In a plane, the interior of a circle consists of the points that are inside the circle. The exterior of a circle consists of the points that are outside the circle. Properties of Tangents. The point at which a tangent line intersects the circle to which it is tangent is the point of tangency.
Following are a few theorems involving points of tangency, and the proofs of each of these theorems. Proof of 1 ST Theorem:. Proof of 2 ND Theorem:. Materials: Compass, Patty Paper, Straightedge. Step 1: Use a compass to draw a circle on patty paper. Connect and share knowledge within a single location that is structured and easy to search.
I've tried drawing a parallel chord to the tangent but then how would you prove that the chord is perpendicular to the radius? This follows from Hilbert's first axiom of congruence for line segments. Therefore, angle OAQ is 90 degrees, because, straight lines whose slopes are negative reciprocals of each other are perpendicular. The following shows how a chord between separate real root intersections become a single real repeated root at tangent point.
There are two roots. For point of tangency there should be a double root or a single coincident root. Basically, since the tangent is perpendicular to the radius drawn to the point of contact and the chord is parallel to the tangent, thus the chord is perpendicular to the radius drawn to the point of contact.
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