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A rectangle ABCD exists such that AB = 6 units and BC = 12 units. E and F trisect the diagonal AC in 3 equal parts. G is centroid of the triangle formed by DEF. What is the area of DGBC?
- a)60 sq units
- b)36 sq units
- c)24 sq units
- d)48 sq units
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
A rectangle ABCD exists such that AB = 6 units and BC = 12 units. E an...
Let us draw a rectangle on the coordinate axis. With (0,0) be one of the edge A. B can be (6,0), C(6,12) and D(0,12). E and F divide the AC in 3 equal part. E has to be (2,4) and F (4,8)
G is centroid of DEF, Coordinates of G will be
Upon finding G the required diagram looks like this
Area DGBC = Area GBC + Area DCG
Area DGBC =
Area DGBC = 24+12 = 36 sq units
G is centroid of DEF, Coordinates of G will be
Upon finding G the required diagram looks like this
Area DGBC = Area GBC + Area DCG
Area DGBC =
Area DGBC = 24+12 = 36 sq units
Most Upvoted Answer
A rectangle ABCD exists such that AB = 6 units and BC = 12 units. E an...
To find the area of the quadrilateral DGBC, we can break it down into two triangles, namely DGC and GBC. Let's calculate the area of each triangle separately and then add them to find the total area of DGBC.
Finding the Area of Triangle DGC:
Since G is the centroid of triangle DEF, we know that the line segment DG is divided by G into two parts in a 2:1 ratio. Let's assume the length of DG is x units. This means that GC will be 2x units.
Since the centroid divides the median in a 2:1 ratio, we can also say that the length of GF is 2x units and EF is 4x units.
We know that the length of AC is 12 units, and it is divided into three equal parts by E and F. Therefore, EF + GF = 12 units.
4x + 2x = 12 units
6x = 12 units
x = 2 units
Now that we know the length of DG is 2 units, we can calculate the length of GC as 2x = 4 units.
To find the area of triangle DGC, we can use the formula for the area of a triangle: Area = (1/2) * base * height.
The base of triangle DGC is GC, which is 4 units, and the height is DG, which is 2 units. Plugging these values into the formula, we get:
Area DGC = (1/2) * 4 units * 2 units = 4 square units.
Finding the Area of Triangle GBC:
Since G is the centroid of triangle DEF, we know that the line segment GC is divided by G into two parts in a 2:1 ratio. Let's assume the length of GC is y units. This means that GB will be 2y units.
Since the centroid divides the median in a 2:1 ratio, we can also say that the length of GF is y units and EF is 2y units.
Using the fact that EF + GF = 12 units, we can substitute the values and solve for y:
2y + y = 12 units
3y = 12 units
y = 4 units
Now that we know the length of GC is 4 units, we can calculate the length of GB as 2y = 8 units.
To find the area of triangle GBC, we can use the formula for the area of a triangle: Area = (1/2) * base * height.
The base of triangle GBC is BC, which is 12 units, and the height is GB, which is 8 units. Plugging these values into the formula, we get:
Area GBC = (1/2) * 12 units * 8 units = 48 square units.
Finding the Total Area of DGBC:
To find the total area of DGBC, we need to add the areas of triangles DGC and GBC:
Total Area DGBC = Area DGC + Area GBC
Total Area DGBC = 4 square units + 48 square units
Total Area DGBC = 52 square units
Therefore, the correct answer is option D) 52 square units.
Finding the Area of Triangle DGC:
Since G is the centroid of triangle DEF, we know that the line segment DG is divided by G into two parts in a 2:1 ratio. Let's assume the length of DG is x units. This means that GC will be 2x units.
Since the centroid divides the median in a 2:1 ratio, we can also say that the length of GF is 2x units and EF is 4x units.
We know that the length of AC is 12 units, and it is divided into three equal parts by E and F. Therefore, EF + GF = 12 units.
4x + 2x = 12 units
6x = 12 units
x = 2 units
Now that we know the length of DG is 2 units, we can calculate the length of GC as 2x = 4 units.
To find the area of triangle DGC, we can use the formula for the area of a triangle: Area = (1/2) * base * height.
The base of triangle DGC is GC, which is 4 units, and the height is DG, which is 2 units. Plugging these values into the formula, we get:
Area DGC = (1/2) * 4 units * 2 units = 4 square units.
Finding the Area of Triangle GBC:
Since G is the centroid of triangle DEF, we know that the line segment GC is divided by G into two parts in a 2:1 ratio. Let's assume the length of GC is y units. This means that GB will be 2y units.
Since the centroid divides the median in a 2:1 ratio, we can also say that the length of GF is y units and EF is 2y units.
Using the fact that EF + GF = 12 units, we can substitute the values and solve for y:
2y + y = 12 units
3y = 12 units
y = 4 units
Now that we know the length of GC is 4 units, we can calculate the length of GB as 2y = 8 units.
To find the area of triangle GBC, we can use the formula for the area of a triangle: Area = (1/2) * base * height.
The base of triangle GBC is BC, which is 12 units, and the height is GB, which is 8 units. Plugging these values into the formula, we get:
Area GBC = (1/2) * 12 units * 8 units = 48 square units.
Finding the Total Area of DGBC:
To find the total area of DGBC, we need to add the areas of triangles DGC and GBC:
Total Area DGBC = Area DGC + Area GBC
Total Area DGBC = 4 square units + 48 square units
Total Area DGBC = 52 square units
Therefore, the correct answer is option D) 52 square units.
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A rectangle ABCD exists such that AB = 6 units and BC = 12 units. E and F trisect the diagonal AC in 3 equal parts. G is centroid of the triangle formed by DEF. What is the area of DGBC?a)60 sq unitsb)36 sq unitsc)24 sq unitsd)48 sq unitsCorrect answer is option 'B'. Can you explain this answer?
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