Area Triangle Coordinate Geometry 

Coordinate geometry is the study of geometry by looking at the coordinate points on the plane at any length. coordinate geometry is used to find the distance between two points, divide lines into ratios, find the midpoint of a line, figure out the area of a triangle, and more. You can use many different ways to figure out how big the area of a triangle is based on what you know about it. For example, you can figure out how big the base and height are, how far apart the sides are, and how many vertices it has. There are three sides to a triangle in Geometry. It is also called a polygon that has three edges and three points. The area of the triangle is how much space the triangle takes up in a two-dimensional plane. The formula to find the area of a triangle is 1/2 x base x height.

Coordinate geometry 

It is the study of geometry with coordinate points. The area of a triangle in coordinate geometry can be found if the three points on each side are known. This is called “geometry.” Area in coordinate geometry is how much space or area a triangle covers in the 2-D coordinate plane. This is how you measure it.

Formula 

Area of ∆ABC = ½ [ x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)] 

Derivation of formula 

Area of ∆ ABC = area of trapezium ABQP + area of trapezium APRC – area of trapezium BQRC.

We know that area of trapezium = ½ (sum of parallel sides) (distance between them) 

Area of ∆ ABC = ½ (BQ + AP) QP + ½ (AP + CR) PR – ½ (BQ + CR) QR

= ½ (y2 + y1) (x1 – x2) + ½ (y1 + y3) (x3 – x1) – ½ (y2 + y3) (x3 – x2)

= ½ [ x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)]

= thus, the area of ∆ABC is the numeric value of the expression 

½ [ x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)] 

Area of segment of the circle 

When a circle is segmented, it refers to the part of the circle that is bordered by an arc and a chord. In the case of a division into parts, each portion is referred to as a segment in its own right. Similar to this, a segment is considered to be a portion of a circle. A segment, on the other hand, is not just any random section of a circle; rather, it is a definite part of a circle that is cut by a chord that runs through it.

Properties of a segment of a circle 

1. It is the region that is bounded by a chord and an arc, respectively.

2. The angle subtended by the segment at the center of the circle is the same as the angle subtended by the arc that intersects the segment at the center. This angle is referred to as the central angle in most cases.

3. A minor segment is formed by subtracting the area of the circle occupied by the corresponding major segment from the total area of the circle.

4. A major segment is obtained by subtracting the area of the circle’s total area from the area of the matching minor segment.

5. An equilateral semicircle is the biggest section of any circle created by its diameter and the arc that connects the two halves.

How to calculate the area of a segment of a circle?

Area of a segment of a circle = area of the sector – area of triangle 

= 0.5 x r2 x a – 0.5 x r2 sin (a) 

= 0.5 x r2 x (a- sin(a)) 

Some solved questions 

Q1. Find the area of a triangle formed by the points A(5, 2), B(4, 7), and C (7, – 4).

Solution –  Given –  A(5, 2), B(4, 7), and C (7, – 4)

Area of ∆ABC= ½ [ x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)] 

= ½ [ 5 (7 + 4) + 4 (-4 – 2) + 7 (2 – 7)] 

½ (55-24-35) = -4/2 = -2.

Q2. Find the area of a triangle whose vertices are (1,-1) (-4,6), and (-3,-5).

Answer – we will solve this question by using the formula,

Area of triangle ABC = ½ [ x1 (y2-y3) + x2 (y3-y1) + x3(y1-y2)]

Let x1= 1, y1= -1 

X2 = -4, y2= 6 

X3= -3, y3= -5

On substituting these values in formula we get,

½ [ 1 (6- (-5)) + ((-4) (-5-(-1)) + (-3) (-1-6)]

½ [ 1 (6+5) +(-4) (-5+1) + (-3) (-7)]

½  [ 11 + 16 + 21 ] 

½ [48] 

=24 square units.

Practice questions 

1. Find the area of the triangle whose vertices are: (i) (2, 3), (–1, 0), (2, – 4) (ii) (–5, –1), (3, –5), (5, 2) 2. In each of the following find the value of ‘k’, for which the points are collinear. (i) (7, –2), (5, 1), (3, k) (ii) (8, 1), (k, – 4), (2, –5)

3. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, –1), (2, 1), and (0, 3). Find the ratio of this area to the area of the given triangle.

4. Find the area of the quadrilateral whose vertices, taken in order, are (– 4, – 2), (– 3, – 5), (3, – 2), and (2, 3).

Conclusion 

Coordinate geometry is the study of geometry with coordinate points. Area in coordinate geometry is how much space or area a triangle covers in the 2-D coordinate plane. The formula to find the area of a triangle is 1/2 x base x height. This is how you measure it. A segment is not just any random section of a circle; rather, it is a definite part that is cut by a chord.

An equilateral semicircle is the biggest section of any circle created by its diameter and the arc that connects the two halves. Area of triangle ABC = ½. [x1 (y2 – y3)+ x2 (y3 – y1)+ x3(y1-y2)].