Determine the area of the triangle MNO whose midpoints of the sides MN, NO and OM are P(2, -1), Q (4,2) and R(1,-4) respectively. s: A. O 54 sq units B. O 6 sq units C. O 42 sq units D. O 2 sq units
Answers
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Given:
Midpoints of sides of the triangle: P(2, -1), Q (4,2) and R(1,-4)
To find:
The area of triangle MNO
Solution:
The area of triangle MNO is 6 sq units. (Option B)
We can find the area by following the given steps-
We know that the area of the triangle formed by joining the midpoints of sides of a triangle is 1/4th of the area of the triangle.
So, the area of triangle PQR=1/4th of the area of triangle MNO.
We are given that the coordinates of the midpoints are P(2, -1), Q (4,2) and R(1,-4).
Using coordinate geometry, the area of a triangle=1/2[x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]
The coordinates of P= (x1, y1)= (2, -1)
The coordinates of Q= (x2, y2)= (4, 2)
The coordinates of R= (x3, y3)= (1, -4)
Now we will put the values in the formula,
Area of ΔPQR=1/2× |[2(2-(-4))+4(-4-(-1))+1(-1-2)]|
=1/2×|[2(6)+4(-3)+1(-3)]|
=1/2×|(12-12-3)|
=1/2×|(-3)|
= 3/2 sq units
We know that this area is 1/4th of the area of ΔMNO.
So, the area of triangle MNO=4×area of triangle PQR
On putting the values, we get
Area of ΔMNO=4×(3/2)
= 6 sq units
Therefore, the area of triangle MNO is 6 sq units.