In Geometry, a triangle is a three-sided polygon that has three edges and three vertices. The area of triangle in coordinate geometry is a measure of the space covered by the triangle in the two-dimensional plane. If the three vertices of the triangle are given in the coordinate plane, the area can be determined. In this article, let us discuss what the area of a triangle is, various methods to find the area of the triangle according to the given parameters and formula of area of triangle in coordinate geometry along with solved examples.
Formula of Area of Triangle in Coordinate Geometry
There are three methods for calculating the area of a triangle. The three approaches are described in detail below.
Method 1
When the triangle's base and altitude are known.
Area of the triangle, A=\[\frac{bh}{2}\]square units
Where b and h are the triangle's base and altitude, respectively.
Method 2
When the length of three sides of the triangle is given, then Heron’s Formula is used to calculate the area of a triangle.
Therefore, the equation used to calculate the triangle's area is given below.
A=\[\sqrt{s(s-a)(s-b)(s-c)}\]
The side lengths of the triangle are a, b, and c, and the semi perimeter is s.
The s value is found by using the formula S=\[\frac{a+b+c}{2}\]
Area of Triangle with Coordinates
Method 3
In this method, we will learn a formula to calculate the area of triangle with coordinates.
If coordinates points are A(x1,y1),B(x2,y2),C(x3,y3) then the area of triangle using coordinates is $\frac{1}{2}$ [x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]
Derivation of the Formula is given below:
Assume a triangle PQR with the coordinates P, Q, and R given as (x1,y1),(x2,y2),(x3,y3)
respectively.
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From the figure, the area of a triangle PQR, lines such as \[\underset{QA}{\leftrightarrow},\underset{PB}{\leftrightarrow}And\underset{RC}{\leftrightarrow}\] are drawn from Q, P and R, respectively perpendicular to the x-axis.
In the coordinate plane, three separate trapeziums, such as PQAB, PBCR, and QACR, are now created.
Calculate the region of each trapezium now.
Therefore, the area of ∆PQR can be calculated as Area of ∆PQR=[\text{Area of trapezium PQAB + Area of trapezium PBCR}] -[\text{Area of trapezium QACR}] —-(1)
Finding Area of a Trapezium PQAB
The formula for calculating the region of a trapezium is as follows:
Since Area of a trapezium = $\frac{1}{2}$ (sum of the parallel sides)×(distance between them)
Area of trapezium PQAB = $\frac{1}{2}$ (QA + PB) × AB
QA = y2
PB = y1
AB = OB – OA = x1-x2
Area of trapezium PQAB = $\frac{1}{2}$ (y1+y2)(x1-x2) —-(2)
Finding Area of a Trapezium PBCR
Area of trapezium PBCR = $\frac{1}{2}$ (PB + CR) × BC
PB = y1
CR =y3
BC = OC – OB = x3-x1
Area of trapezium PBCR = $\frac{1}{2}$ (y1+y3)(x3-x1) (1/2) —-(3)
Finding Area of a Trapezium QACR
Area of trapezium QACR = $\frac{1}{2}$ (QA + CR) × AC
QA = y2
CR = y3
AC = OC – OA = x3-x2
Area of trapezium QACR = $\frac{1}{2}$ (y2+y3)(x3-x2)-(4)
Substituting (2), (3) and (4) in (1),
Area of ∆PQR = $\frac{1}{2}$ [(y1+y2)(x1-x2)+(y1+y3)(x3-x1)-(y2+y3)(x3-x2)
A=$\frac{1}{2}$ [x1(y2-y3)+x2(y3-y1)+x3(y1-y2)
Special Case:
If one of the triangle's vertices is the origin, the triangle's area can be determined using the formula below.
Calculate area of a triangle with vertices are (0,0), P(a, b), and Q(c, d)
A = $\frac{1}{2}$ [0(b – d) + a(d – 0) + c(0 – b)]
A = (ad – bc)/2
Area of Triangle with Vertices
When the area of a triangle with vertices P(x1,y1),Q(x2,y2) and R(x3,y3) is zero, then $\frac{1}{2}$ [x1(y2-y3)+x2(y3-y1)+x3(y1-y2)=0 and the points P(x1,y1),Q(x2,y2) and R(x3,y3)are collinear. They do not form a triangle since they are in a straight line.
Area of triangle with 3 points
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The formula for calculating area can be expressed using determinants, as shown below.
\[\alpha =\frac{1}{2}\begin{vmatrix}
X_{1} &Y_{1} &1 \\
X_{2}&Y_{2} & 1\\
X_{3} &Y_{3} & 1
\end{vmatrix}\]
Where denotes the area of a triangle and A(x1,y1),B(x2,y2) and C(x3,y3)represent the vertices of the triangle.
We know the determinant's value can be either negative or positive. Since we are finding an area and it can never be taken as a negative, therefore we should take the absolute value of the determinant.
We use both the positive and negative values of the determinant if the triangle's area is already known.
Area of triangle with 3 points formula is \[\alpha =\frac{1}{2}\][x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]
Another form
The following pictorial representation makes writing the above formula very simple.
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Area of ABC=12{(x1.y2+x2y3+x3y1)-(x2y1+x3y2+x1y3) sq. units.
Note: As we know a triangle's area can never be negative. So we must take the absolute value, in case the area happens to be negative.
Area of a Triangle in Coordinate Geometry Example
1. Calculate the area of the ∆ABC whose vertices are A(1, 2), B(4, 2) and C(3, 5)?
Sol: Using the formula,
12[x1(y2-y3)+x2(y3-y1)+x3(y1-y2)
A=12[1(2-5)+4(5-2)+3(2-2)]
A=12[-3+12]
A=92 square units.
Hence, the area of a triangle ABC is 92 square units.
2. Find the area of the triangle using vertices A(3,4), B(4,7) and C(3,5).
Sol: Here x1=3,y1=4,x2=4,y2=7,x3=3 and y3=5
We know the formula of area of a triangle is given by
12[x1(y2-y3)+x2(y3-y1)+x3(y1-y2)
=12[3(7-3)+4((-3)-(-4))+6(4-(-7))]
=1230+4-18
=8 sq. units
Hence, the area of a triangle is 8 unit square
Conclusion:
We have discussed the area of triangle formula in coordinate geometry. When three points are collinear, the triangle's area is zero. The area of a triangle cannot be negative. When we get the answer in negative terms, we should consider the numerical value of the area, without the negative sign as the area is always positive.
FAQs on Area Of Triangle In Coordinate Geometry
1. What is the standard formula to calculate the area of a triangle using the coordinates of its vertices?
To find the area of a triangle with vertices at the points A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), you can use the following formula:
Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
Here, (x₁, y₁), (x₂, y₂), and (x₃, y₃) are the coordinates of the three vertices of the triangle. The absolute value is taken because the area must always be a positive quantity.
2. What is the relationship between the area of a triangle and the collinearity of its three vertices?
The relationship is a foundational concept in coordinate geometry. If the area of a triangle calculated using the coordinates of its three vertices is zero, it means the points do not form a triangle. Instead, they all lie on the same straight line, and are therefore collinear. This is a common method to test if three given points are collinear.
3. How can you find the area of a triangle using the determinant method?
The determinant method is a shortcut for applying the standard area formula. The area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) can be expressed as half the absolute value of the following determinant:
Area = ½ |det(A)| where A is the matrix:
| x₁ y₁ 1 |
| x₂ y₂ 1 |
| x₃ y₃ 1 |
Expanding this determinant gives the same expression as the standard formula. It is often considered a more systematic way to remember and apply the calculation.
4. Why does the area of a triangle formula sometimes result in a negative value, and what does it signify?
The formula for the area of a triangle can yield a positive or negative value depending on the order in which the vertices are taken (clockwise or counter-clockwise). However, since area is a physical quantity that cannot be negative, you must always take the absolute value (or the numerical value without the negative sign) of the final result. The sign itself indicates the orientation of the vertices, but for calculating area, only the magnitude matters.
5. How can you apply the coordinate geometry formula for a triangle's area to find the area of a quadrilateral?
To find the area of a quadrilateral with four given vertices, you can use the triangle area formula with this simple method:
- Step 1: Divide the quadrilateral into two triangles by drawing a diagonal between any two opposite vertices.
- Step 2: Calculate the area of the first triangle using the coordinates of its three vertices.
- Step 3: Calculate the area of the second triangle similarly.
- Step 4: Add the areas of the two triangles to get the total area of the quadrilateral.
6. What is the geometric logic behind the coordinate formula for the area of a triangle?
The formula is derived by a clever geometric construction. It involves plotting the triangle's vertices on a Cartesian plane and drawing perpendicular lines from each vertex to the x-axis. This creates several trapeziums. The area of the triangle is then calculated by strategically adding and subtracting the areas of these trapeziums to isolate the area enclosed by the triangle's vertices. This method provides a visual proof for the algebraic formula.
7. Is there a simpler or shortcut formula to find the area if one vertex of the triangle is at the origin (0, 0)?
Yes, if one vertex is at the origin (0, 0) and the other two vertices are at (x₁, y₁) and (x₂, y₂), the formula simplifies significantly. The area can be calculated directly using:
Area = ½ |x₁y₂ − x₂y₁|
This is a special case of the general formula and is very useful for quick calculations in such scenarios.
