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If AD is the median of the triangle ABC and G be the centroid, then the ratio of AG:AD is
- a)2:1
- b)3:2
- c)2:3
- d)1:3
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If AD is the median of the triangle ABC and G be the centroid, then th...
If AD is the median of the triangle ABC and G be the centroid then we know that
AG:GD = 2:1
Now adding the numerator in the denominator we get,
⇒ AG : (AG + GD) = 2: (2 +1)
⇒ AG : AD = 2:3
AG:GD = 2:1
Now adding the numerator in the denominator we get,
⇒ AG : (AG + GD) = 2: (2 +1)
⇒ AG : AD = 2:3
Most Upvoted Answer
If AD is the median of the triangle ABC and G be the centroid, then th...
To understand why the ratio of AG:AD is 2:3, let's break down the properties of a median and a centroid in a triangle.
1. Median:
A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In this case, AD is the median of triangle ABC, which means it connects vertex A to the midpoint of side BC.
2. Centroid:
The centroid of a triangle is the point of concurrency of the three medians. In other words, it is the point where all three medians intersect. Let's denote this point as G.
Now, let's analyze the properties of the centroid:
- The centroid divides each median into two segments. The segment closer to the vertex is twice as long as the segment closer to the midpoint of the opposite side.
- The centroid divides each median in a 2:1 ratio.
Now, let's apply these properties to the given triangle ABC and find the ratio of AG:AD:
1. Identify the midpoint of side BC:
Let's say that the midpoint of side BC is denoted as M.
2. Connect vertex A to point M:
Draw a line segment AM.
3. Identify the point of intersection of medians:
The point of intersection of medians is the centroid G.
4. Find the length of AG:
Since G is the centroid, AG is one of the medians. According to the centroid properties, AG can be divided into two segments in a 2:1 ratio. Let's denote the shorter segment as x. Then, the longer segment would be 2x.
Therefore, AG = x + 2x = 3x.
5. Find the length of AD:
AD is the median of the triangle, which connects vertex A to the midpoint of side BC. Therefore, AD is equal to the length of AM.
6. Determine the ratio of AG:AD:
Since AG = 3x and AD = AM, the ratio of AG:AD can be expressed as 3x:AM.
7. Determine the relationship between x and AM:
Since AM is the midpoint of side BC, it divides BC into two equal segments. Therefore, AM is half the length of BC.
Let's denote the length of BC as 2y. Then, AM = y.
8. Substitute the values in the ratio:
Substituting the values, the ratio of AG:AD becomes 3x:y.
9. Determine the relationship between x and y:
Since AD is the median, it divides BC into two equal segments. Therefore, BD = DC = y.
From triangle ABD, we can see that AD is equal to 2x.
Therefore, 2x = y.
10. Substitute the value of y in the ratio:
Substituting the value of y, the ratio of AG:AD becomes 3x:2x.
11. Simplify the ratio:
Canceling out the common factor of x, the ratio simplifies to 3:2.
Hence, the correct ratio of AG:AD is 3:2, which is option 'C'.
1. Median:
A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In this case, AD is the median of triangle ABC, which means it connects vertex A to the midpoint of side BC.
2. Centroid:
The centroid of a triangle is the point of concurrency of the three medians. In other words, it is the point where all three medians intersect. Let's denote this point as G.
Now, let's analyze the properties of the centroid:
- The centroid divides each median into two segments. The segment closer to the vertex is twice as long as the segment closer to the midpoint of the opposite side.
- The centroid divides each median in a 2:1 ratio.
Now, let's apply these properties to the given triangle ABC and find the ratio of AG:AD:
1. Identify the midpoint of side BC:
Let's say that the midpoint of side BC is denoted as M.
2. Connect vertex A to point M:
Draw a line segment AM.
3. Identify the point of intersection of medians:
The point of intersection of medians is the centroid G.
4. Find the length of AG:
Since G is the centroid, AG is one of the medians. According to the centroid properties, AG can be divided into two segments in a 2:1 ratio. Let's denote the shorter segment as x. Then, the longer segment would be 2x.
Therefore, AG = x + 2x = 3x.
5. Find the length of AD:
AD is the median of the triangle, which connects vertex A to the midpoint of side BC. Therefore, AD is equal to the length of AM.
6. Determine the ratio of AG:AD:
Since AG = 3x and AD = AM, the ratio of AG:AD can be expressed as 3x:AM.
7. Determine the relationship between x and AM:
Since AM is the midpoint of side BC, it divides BC into two equal segments. Therefore, AM is half the length of BC.
Let's denote the length of BC as 2y. Then, AM = y.
8. Substitute the values in the ratio:
Substituting the values, the ratio of AG:AD becomes 3x:y.
9. Determine the relationship between x and y:
Since AD is the median, it divides BC into two equal segments. Therefore, BD = DC = y.
From triangle ABD, we can see that AD is equal to 2x.
Therefore, 2x = y.
10. Substitute the value of y in the ratio:
Substituting the value of y, the ratio of AG:AD becomes 3x:2x.
11. Simplify the ratio:
Canceling out the common factor of x, the ratio simplifies to 3:2.
Hence, the correct ratio of AG:AD is 3:2, which is option 'C'.
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If AD is the median of the triangle ABC and G be the centroid, then the ratio of AG:AD isa)2:1b)3:2c)2:3d)1:3Correct answer is option 'C'. Can you explain this answer?
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If AD is the median of the triangle ABC and G be the centroid, then the ratio of AG:AD isa)2:1b)3:2c)2:3d)1:3Correct answer is option 'C'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about If AD is the median of the triangle ABC and G be the centroid, then the ratio of AG:AD isa)2:1b)3:2c)2:3d)1:3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If AD is the median of the triangle ABC and G be the centroid, then the ratio of AG:AD isa)2:1b)3:2c)2:3d)1:3Correct answer is option 'C'. Can you explain this answer?.
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