The vertices of a ΔABC are A(4,6), B(1,5) and C(7,2). A line...

Key Concepts

Coordinate Geometry

Coordinate geometry involves representing geometric figures using ordered pairs in the Cartesian plane, enabling the application of algebraic techniques to solve geometric problems. It allows us to calculate distances, slopes, and areas based on vertices’ coordinates, which is essential for analyzing triangles and other polygons.

Section Formula

The section formula is a tool used to locate a point that divides a line segment into a given ratio. In problems where points are determined by proportional distances along the sides of a triangle, this formula provides the means to find their exact coordinates, facilitating further calculations.

Similarity and Area Ratios

When triangles are similar, the ratios of corresponding sides are equal and the areas scale with the square of the similarity ratio. This concept is crucial in determining how the area of a smaller, similar triangle (formed by dividing sides proportionally) relates to the area of the original triangle.

Determinant Method for Calculating Area

The determinant method is a formula used to compute the area of a triangle when the coordinates of its vertices are known. It involves forming a matrix from the coordinates of the vertices and calculating half the absolute value of its determinant, providing an efficient algebraic way to find the area.

Transcript

00:01 Hello students, welcome back.

00:02 In this question, we are given a triangle, triangle a, b, c, and we are given the vertices of this triangle.

00:11 Okay, a is given to be 4 .6, b is given to be 1 .5.

00:18 C is given to be 7 .1.

00:23 Okay.

00:24 So now in this question, we are given, sorry, it's 7 .2.

00:30 Okay.

00:31 So, and this question we are given, sorry, it's 7 .2.

00:32 Are given the vertices of this triangle ab c are this ab and c a line is drawn to intersect ab and ac at d and e respectively such that ad upon ab is equal to a e equal to a c is equal to four so we are given a condition that ad upon ab is equals to a e upon ac which is equals to one by four okay so we are given this condition now we have to calculate the area of the triangle a d e and then we have to compare it with the area of triangle abc okay so since we are given this condition i'll just split this equation up okay so let me draw this point b and e point okay so let's say this is point d and let's say this is point e okay so if i if i rewrite the given equation in a different form what do i get is a d divided by d b is equal to 1 by 3 right which is equal to a e divided by ec right because what we are given is that a d this part is one fourth of the ab okay since this uh this small part is one fourth of this whole part then what we can say that this part is one third of this part, right? so i have written ad divided by db is equal to 1 by 3.

02:13 Now we have to find out the vertices of d, okay, the coordinates of d.

02:19 So for d, x will be equal to mx2 plus nx1 divided by m plus n.

02:28 Okay, i have applied section formula.

02:30 So if i put the values, what do i get? get is 1 multiplied by 1 plus 3 multiplied by 4 divided by 3 plus 1.

02:41 So the x coordinate of d comes out to be 13 by 4.

02:45 Similarly, finding out the y coordinate, what do i get is my2 plus n y1 divided by m plus n...

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