22. A (4,3), B (6,5) and C (5,-2) are the vertices of triangle ABC. (i) Find the co-ordinates of the centroid G of triangle ABC. Find the area of triangle GBC and compare it with area of triangle ABC. If D is the mid-point of BC, find the co-ordinates of D. Find the co-ordinates of a point P on AD such that AP : PD = 2:3. Find the area of triangle PBC and compare it with area of triangle ABC.
Answers
Step-by-step explanation:
Hint: We will use the very important concept that if we have a triangle ABC and D, E and F are the midpoints of BC, CA and AB respectively, then centroid of triangle ABC coincides with the centroid of triangle DEF by using the centroid formula which is as follows:
If (x1,y1),(x2,y2),(x3,y3)
are the vertices of triangle, then the centroid of triangle is (x1+x2+x33,y1+y2+y33)
Complete step-by-step answer:
We have been given the midpoint of sides of the triangle as D (-7, 6), E (5, 8) and F (2, -2).
Let us suppose the triangle to be ABC and D, E and F are the midpoints of BC, AC and AB.
Now we know that if we have any triangle ABC and their midpoint’s area D, E and F then the centroid of triangle ABC coincides with the centroid of triangle DEF.
So we will find the centroid of triangle DEF which gives the centroid of triangle ABC.
We know that if (x1,y1),(x2,y2),(x3,y3)
are the vertices of the triangle, then,
X coordinate of centroid =x1+x2+x33
Y coordinate of centroid =y1+y2+y33
We have D (-7, 6), E (5, 8) and F (2, -2)
X coordinate of centroid of triangle DEF =−7+5+23=03=0
Y coordinate of centroid of triangle DEF =6+8+(−2)3=14−23=123=4
So the coordinates of the centroid of the triangle DEF are (0, 4)
Hence the coordinates of the centroid of the triangle ABC are (0, 4).
Note: Be careful while finding the values of the coordinates of the centroid and also take care of the sign while substituting the values of coordinates of the given midpoints. Also, remember that the centroid of a triangle is a point where all the three medians of the triangle intersect