Mathematics : Coordinate Geometry: Area of a Triangle and Quadrilateral: Solved Example Problems

Find the area of the triangle whose vertices are** **(-3, 5)** **,** **(5, 6)** **and** **(5,** **-** **2)

Plot the points in a rough diagram and take them in
counter-clockwise order.

Show that the points *P*(-1.5, 3) , *Q*(6 , -2) , *R*(-3
, 4) are collinear.

The points are *P*(-1. 5, 3) , *Q*(6 , -2) , *R*(-3
, 4)

Area of Δ*PQR *= 1/2{(*x* _{1}*y*_{2}
+ *x* _{2}*y*_{3} + *x* _{3}*y*_{1}
) −(*x* _{2}*y*_{1} + *x* _{3}*y*_{2}
+ *x* _{1}*y*_{3} )}

=1/2 {(3 + 24 − 9) −(18 + 6 − 6)} = 1/2{18 −18} = 0

Therefore, the given points are collinear.

**Example 5.3**

If the area of the triangle formed by the vertices *A*(-1, 2)
, *B* (*k* , -2) and *C*(7, 4)* *(taken in order) is 22 sq.
units, find the value of* k*.

*Solution*

The vertices are *A*(-1, 2) , *B* (*k* , -2) and *C*(7,
4)

Area of triangle *ABC* is 22 sq.units

2*k* + 34 = 44 gives 2*k* = 10 so *k* = 5

**Example 5.4**

If the points *P*(- 1, -4) , *Q* (*b*,*c*) and
*R*(5, -1) are collinear and if 2*b* + *c* = 4 , then find the
values of *b* and *c.*

* Solution*

Since the three points *P*(- 1, - 4) , *Q* (*b*,*c*)
and *R*(5, -1) are collinear,

* *Area of triangle *PQR *= 0

−*c* −*b* − 20 + 4*b* − 5*c* −1 = 0

*b *-* *2*c *=* *7 …(1)

Also, 2*b* + *c* = 4 …(2) (from given information)

Solving (1) and (2) we get *b* = 3 , *c* = −2

The floor of a hall is covered with identical tiles which are in
the shapes of** **triangles. One such triangle has the vertices at (-3 , 2) , (- 1 ,
-1) and (1 , 2) . If the floor of the hall is completely covered by 110 tiles,
find the area of the floor.

Vertices of one triangular tile are at

(-3 , 2) , (- 1 , -1) and (1 , 2)

Area of this tile = 1/2 {(3 − 2 + 2) −(− 2 −1 − 6)} sq.units

= 1/2(12) = 6 sq.units

Since the floor is covered by 110 triangle shaped identical tiles,

Area of floor = 110 ×6 = 660 sq.units

Find the area of the quadrilateral formed by the points** **(8, 6)** **,** **(5, 11)** **,** **(-5, 12) and (-4, 3) .

Before determining the area of quadrilateral, plot the vertices in
a graph. Let the vertices be *A*(8,6), *B*(5,11), *C*(–5,12) and
*D*(–4,3)

Therefore, area of the quadrilateral *ABCD*

The given diagram shows a plan for constructing a new parking lot at
a campus. It is estimated that such construction would cost ₹1300 per square
feet. What will be the total cost for making the parking lot?

The parking lot is a quadrilateral whose vertices are at A(2,
2) , B(5, 5) , C(4, 9) and D(1, 7) .

Therefore, Area of parking lot

So, area of parking lot = 16 sq.feets

Construction rate per square feet = ₹1300

Therefore, total cost for constructing the parking lot = 16 ×1300
= ₹20800

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