Q: Find the area of a triangle whose vertices are $(1, -1), (-4, 6)$ and $(-3, -5)$. Also, derive and explain the formula used to calculate the area of a triangle in coordinate geometry.

Introduction to Area of a Triangle in Coordinate Geometry\nIn elementary geometry, the area of a triangle is typically calculated using the formula $\frac{1}{2} \times \text{base} \times \text{height}$. However, when only the Cartesian coordinates of the three vertices are provided, determining the

Question 1

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

Accepted Answer

The area of a triangle formed by three vertices $(x1, y1)$, $(x2, y2)$, and $(x3, y3)$ is given by the formula: $$	ext{Area} = \frac{1}{2} |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|$$ 
Given vertices are: $(x1, y1) = (1, -1)$ $(x2, y2) = (-4, 6)$ $(x3, y3) = (-3, -5)$ 
Substitute these values into

Question 2

Determine the ratio in which the line segment joining the points $(-3, 10)$ and $(6, -8)$ is divided by $(-1, 6)$.

Accepted Answer

Let the given points be $A(-3, 10)$ and $B(6, -8)$, and let the dividing point be $P(-1, 6)$. Assume that point $P$ divides the line segment $AB$ in the ratio $k : 1$. 
Using the section formula for the x-coordinate: $$x = \frac{k(x2) + 1(x1)}{k + 1}$$ 
Substitute the given values into the formula

Question 3

Find the relation between $x$ and $y$ such that the point $(x, y)$ is equidistant from the points $(3, 6)$ and $(-3, 4)$.

Accepted Answer

Let the given point be $P(x, y)$, and the two fixed points be $A(3, 6)$ and $B(-3, 4)$. According to the problem, point $P$ is equidistant from $A$ and $B$, which means: $$PA = PB$$ 
Squaring both sides to eliminate the square roots, we get: $$PA^2 = PB^2$$ 
Using the distance formula, we write: $

Question 4

Find the area of a triangle whose vertices are $(1, -1), (-4, 6)$ and $(-3, -5)$. Also, derive and explain the formula used to calculate the area of a triangle in coordinate geometry.

Accepted Answer

Introduction to Area of a Triangle in Coordinate Geometry
In elementary geometry, the area of a triangle is typically calculated using the formula $\frac{1}{2} 	imes 	ext{base} 	imes 	ext{height}$. However, when only the Cartesian coordinates of the three vertices are provided, determining the