-Consider the parallelogram with vertices at (0,0),(2,0) (3, 2) and (1, 2). Find the angle at wh

-Consider the parallelogram with vertices at (0,0),(2,0) (3,...

Key Concepts

Angle Between Vectors

The angle between two vectors can be determined using the dot product formula, which relates the cosine of the angle to the ratio of the dot product of the vectors to the product of their magnitudes. This concept is widely applied in problems involving the calculation of angles in various geometric contexts, ensuring a rigorous method for quantifying the relationship between vector directions.

Diagonal Intersection

In any parallelogram, the point where the diagonals intersect is the midpoint of each diagonal. Understanding this property is crucial when analyzing relationships within the shape. Even though the diagonals do not generally have the same length, knowing they bisect each other assists in deriving their vector forms and using symmetry to compute the intersection angle.

Parallelogram Properties

A parallelogram is a quadrilateral with two pairs of parallel sides. An essential property of parallelograms is that their opposite sides are congruent and parallel. Additionally, the diagonals of a parallelogram bisect each other, meaning that they intersect at their midpoints, which is key to many geometric proofs and calculations involving these shapes.

Vector Representation and Operations

Vectors are a fundamental concept in coordinate geometry, enabling the representation of geometric objects like sides and diagonals of shapes. By expressing the diagonals as vectors, one can utilize operations such as the dot product to compute angles between them. This approach is particularly useful when working with coordinates, as it allows for the translation of geometric relations into algebraic formulas.

Transcript

00:01 To explain here the tangent proofs and examples here right so first of all we will know that what is tangent actually so tangent to a circle is a straight line that touches the circle at one point right at any one point you can say or only one point to the circle so that point which touches only one point is that point is known as tangency right at tangency point the tangent of the circle will be perpendicular to the radius of the circle a circle can have infinite tangents right so the circle may have let us say this is a circle and this is central o and we are assuming this tangent so this tangent that is straight line will cut at only one point remember this so this one point which is cutting which is by you can say touching the circle is called tangency to the circle right and this perpendicular from dawn any line which is drawn from tangency point from a center then it will be radius and it will be perpendicular to the tangent so this is the basic thing we know about tangent now there could be two tangents draw can be drawn parallel to the secant that can be drawn at the circle right now there would be externally there could be two ways tangents can be drawn for two circles right let us say how first is externally and another is nothing but this is internally let us see how so this is the diagram let us see this this would be the diagram right so this will be one circle and this will have tangent right two external tangents to external tangents right now now it would be one more way this could be this so this will be internally right so if the two circles touch at just one point with with one inside the other there is just one line there is just one line that touch the circle which is tangent i'm sorry i should write here that is that is tangent so here we can see two cases externally there could be two external tangents maximum for two circles and that that touches the two circles externally right for internally it touches only one one you can say one tangent only right so there could be one two more more cases right you can see here in this way this is for case one you can say this is case one here right so now case two what we can do here case two is the condition right here it could be i'm just making the diagrams because it would be so lengthy to explain all the things all together because it will take at least two two classes to explain the tangent right and to take the all the examples right to make it understand i'm only i'm going with very brief examples right so this could be second case right in which you can say a tangent of two circles is a common internal tangent, the intersection of the tangent and the line segment joining the center is not empty, right? for example, line a b, a b, common internal tangent, right? so this has tangency.

06:58 Right.

06:59 So in case three what happens? case 3, tangents of two circles will intersect at a point.

07:08 We can name it as o.

07:10 So, a, b is a common tangent and cd is also a common tangent.

07:14 So these two tangents, a, b, c, d, intersecting at one point.

07:19 So let us see, what is the third case? let us write it.

07:23 Let us draw the diagram, right? so here it will be like this.

07:38 Starting here right so this is a circle again so this will be a c you can say this is b and sorry this is c d tangent right this is okay so this is the case third right so now moving ahead we will just see one proof right so one proof is there theorem we are writing here theorem says the tangent at any point of a circle is perpendicular to the radius all right so we need to prove here opi is perpendicular to a b here right so here is the diagram given to us that is one diagram is here like this this will be this way right so here it is o p this is a and p right so here we can write here o is center of the right first thing now secondly ab is the is the tangent to the circle with the center o.

10:28 Now, p is nothing but p is the point of tangency where tangent intersect with the circle, isn't it? yes.

10:41 So now moving forward let q be the point on the tangent ab, right? which is like this connecting circle center o right so this is q now we are asked that we will draw the imaginary line from point point o to q it touches the it touches at r right so this is r right this is value this point is r so let us write here draw and draw and imaginary line from point o to q it touches the circle at r.

12:15 Right.

12:17 So now as oq is greater than op, right, isn't it? why? because this is kind of perpendicular.

12:27 We know this.

12:28 Or you can say this will be greater because this is still the circle.

12:32 And this is out of the circle right so obviously or will be greater than op so let us write or here o q is greater than o p all right now o q is nothing but this will be equals to o r plus r q so same will be same will be equal with the other points of tangents as well.

13:19 Hence, op is the smallest line connects to the tangents...

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