If (2,3) is one of the vertices and x-y+3=0 is one of the sides of an equilateral triangle, then

If (2,3) is one of the vertices and x-y+3=0 is one of the...

Key Concepts

Equilateral Triangle Properties

An equilateral triangle has all three sides of equal length and all three interior angles equal to 60°. This intrinsic symmetry means that many of its geometric centers—such as the circumcenter, centroid, and orthocenter—coincide, and that rotations by 60° or 120° around its center map the triangle onto itself.

Area of an Equilateral Triangle

The area of an equilateral triangle can be expressed in terms of its side length using the formula A = (?3/4)·a². This result follows from dropping an altitude and using the Pythagorean theorem, linking the side length with the height, and hence allowing computation of the area in a compact form.

Coordinate Geometry and Line Equations

In coordinate geometry, lines are often represented by linear equations which allow us to determine slopes, intercepts, and distances. Understanding line equations such as those of the form ax + by + c = 0 is crucial for analyzing the orientations and intersections of sides in triangles.

Rotation Transformation in the Plane

Rotation transformations involve turning a figure around a point by a certain angle, preserving distances and angles. In the context of triangles, especially equilateral ones, rotating a side or vertex by ±60° helps derive the orientations (or slopes) of the other sides, making it a powerful tool for solving coordinate geometry problems.

Triangle Centers in Equilateral Triangles

Key centers of a triangle include the circumcenter (the center of the circumscribed circle) and the orthocenter (the intersection of the altitudes). In an equilateral triangle, these centers coincide with the centroid due to the symmetry, which simplifies many calculations related to the triangle's geometry.

Transcript

00:01 So in this problem we have been given if 2 comma 3 is one of the vertices and x minus y plus 3 equal to 0 is one of the sides of an equilateral triangle, then we have to choose the right option.

00:11 Now, you can clearly see that the point 2 .3 does not satisfy the line x minus y plus 3 equal to 0.

00:30 So it would not lie on that side.

00:32 Now let me draw the figure, a rough figure of the equilateral triangle.

00:39 So now over here let this be point a and let this point b, 2 .3.

00:45 So therefore this would be point b, point c and this line would be x minus y plus 3 equal to 0.

00:56 Now it is opposite vertex of the given side.

00:59 Hence since a b c is an equilateral triangle, angle b and angle c would be equal to be equal equal to each other of course and we also equal to angle a which is 60 degrees and slopes of ab and ac are given by so let me write it as slopes of ab that is mab and slopes of ac which i'm writing it as mac would be given by tan theta and since theta here is 60 degrees as we have seen above so tan 60 would be equal to m minus 1 upon 1 plus m into 1 all right modulus uh sorry i'll just make this properly yeah this would be since tan theta is formula is tana theta equal to modulus of m1 minus m2 upon 1 plus m1 into m2 so therefore the slopes of mab of ab and ac have been written as mab and and mac and i am taking mab as equal to m and we can clearly see that the slope of the line ac is equal to 1 so therefore we will be getting i'll just write it down the clarification that mab is equal to m and m as equal to 1 and after that we use the formula so therefore m minus 1 by m plus 1 equal to plus or minus root 3 so that would imply m is equal to minus of 2 plus okay minus 2 minus root 3 which would be equal to minus of 2 plus root 3 just actually this is not the one that i am writing just a minute just please ignore this one.

03:26 Yeah.

03:27 Now, once we have done this, therefore we would be getting this this would be equal to m1.

03:40 So let's say that m1 is the slope of ab.

03:44 So slope of ab is equal to m1 which is slope of which would be equal to minus of 2 plus root 3.

03:54 Similarly, we would be getting slope of ac as equal to minus 2 plus root 3 and let this be equal to m2 and let this be equal to m1 so now let us proceed with the first option perpendicular distance from 2 .3 to x minus y plus 3 equal to 0 will be equal to 2 minus 3 plus 3 upon root 2 this is perpendicular distance from the point 2 .3 to x minus y plus 3 equal to 0.

04:31 So this would be equal to root 2.

04:34 Now if a is the length of the side of triangle, then we know that the perpendicular distance root 2 would be equal to root 3 by 2 times of a where a is the length of side of triangle.

04:49 So that would clearly imply a is equal to 2 into root over 2 by 3.

04:58 So therefore area would be equal to we know the formula which is root 3 by 4 a square...

Please give Ace some feedback