
If the coordinates of the points A, B, and C be \[(4,4),(3, - 2),(3, - 16)\] respectively, then find the area of the triangle ABC.
A. 27
B. 15
C. 18
D. 7
Answer
132.9k+ views
Hints First write the formula of the area of a triangle, then substitute the given coordinates in the formula to obtain the required result.
Formula used
Area=\[\dfrac{1}{2}\left[ {{x_1}{\rm{\;}}\left( {{y_{2 - }}{\rm{\;}}{y_3}{\rm{\;}}} \right) + {x_2}{\rm{\;}}\left( {{y_3} - {y_1}{\rm{\;}}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]\] , where \[({x_1},{y_1}),({x_2},{y_2}),({x_3},{y_3})\] are the vertices of the triangle.
Complete step by step solution
Substitute \[({x_1},{y_1}),({x_2},{y_2}),({x_3},{y_3})\] by \[(4,4),(3, - 2),(3, - 16)\] in the formula \[\dfrac{1}{2}\left[ {{x_1}{\rm{\;}}\left( {{y_{2 - }}{\rm{\;}}{y_3}{\rm{\;}}} \right) + {x_2}{\rm{\;}}\left( {{y_3} - {y_1}{\rm{\;}}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]\]and calculate to obtain the required area.
\[\dfrac{1}{2}\left[ {4\left( { - 2 + 16} \right) + 3( - 16 - 4) + 3(4 + 2)} \right]\]
\[ = \dfrac{1}{2}\left[ {56 - 60 + 18} \right]\]
\[ = \dfrac{{14}}{2}\]
=7
The correct option is “D”.
Note While calculating the area of the triangle when the Cartesian coordinates are given, one can also proceed by first plotting the triangle on an X-Y graph. This process can help in identifying the type of triangle that is whether it is an equilateral triangle, isosceles triangle, or right triangle. If we can identify that the triangle is one of them, we can easily calculate the area of the triangle, by using the respective formulas for these special types of triangles. This greatly reduces the time taken in calculating the area of the triangle. If the triangle is not of any special type then use just the general formula. This is also a good approach to doing this type of question.
Formula used
Area=\[\dfrac{1}{2}\left[ {{x_1}{\rm{\;}}\left( {{y_{2 - }}{\rm{\;}}{y_3}{\rm{\;}}} \right) + {x_2}{\rm{\;}}\left( {{y_3} - {y_1}{\rm{\;}}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]\] , where \[({x_1},{y_1}),({x_2},{y_2}),({x_3},{y_3})\] are the vertices of the triangle.
Complete step by step solution
Substitute \[({x_1},{y_1}),({x_2},{y_2}),({x_3},{y_3})\] by \[(4,4),(3, - 2),(3, - 16)\] in the formula \[\dfrac{1}{2}\left[ {{x_1}{\rm{\;}}\left( {{y_{2 - }}{\rm{\;}}{y_3}{\rm{\;}}} \right) + {x_2}{\rm{\;}}\left( {{y_3} - {y_1}{\rm{\;}}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]\]and calculate to obtain the required area.
\[\dfrac{1}{2}\left[ {4\left( { - 2 + 16} \right) + 3( - 16 - 4) + 3(4 + 2)} \right]\]
\[ = \dfrac{1}{2}\left[ {56 - 60 + 18} \right]\]
\[ = \dfrac{{14}}{2}\]
=7
The correct option is “D”.
Note While calculating the area of the triangle when the Cartesian coordinates are given, one can also proceed by first plotting the triangle on an X-Y graph. This process can help in identifying the type of triangle that is whether it is an equilateral triangle, isosceles triangle, or right triangle. If we can identify that the triangle is one of them, we can easily calculate the area of the triangle, by using the respective formulas for these special types of triangles. This greatly reduces the time taken in calculating the area of the triangle. If the triangle is not of any special type then use just the general formula. This is also a good approach to doing this type of question.
Recently Updated Pages
Difference Between Mutually Exclusive and Independent Events

Difference Between Area and Volume

Difference Between Double Salt and Complex Salt: JEE Main 2024

JEE Main 2025: What is the Area of Square Formula?

Difference Between Power and Exponent: JEE Main 2024

Difference Between Pound and Kilogram with Definitions, Relation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility & More

JEE Main Syllabus 2025 (Updated)

JEE Mains 2025 Cutoff: Expected and Category-Wise Qualifying Marks for NITs, IIITs, and GFTIs

JEE Main Marks Vs Percentile Vs Rank 2025: Calculate Percentile Using Marks

How Many Students Will Appear in JEE Main 2025?

NIT Cutoff Percentile for 2025

Other Pages
Maths Question Paper for CBSE Class 10 - 2007

NCERT Solutions for Class 10 Maths Chapter 11 Areas Related To Circles

NCERT Solutions for Class 10 Maths Chapter 12 Surface Area and Volume

NCERT Solutions for Class 10 Maths Chapter 13 Statistics

NCERT Solutions for Class 10 Maths In Hindi Chapter 15 Probability

Areas Related to Circles Class 10 Notes CBSE Maths Chapter 11 (Free PDF Download)
