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D, E, F are the mid points of the sides BC, CA and AB respectively of a triangle ABC. What is the ratio of the circumradii of triangles DEF and ABC respectively?
- a)1 : 4
- b)1 : 2
- c)1 : 8
- d)1 : 16
- e)2 : 3
Correct answer is option 'B'. Can you explain this answer?
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D, E, F are the mid points of the sides BC, CA and AB respectively of ...
where a, b, c are the sides of the triangle and A is the area of the triangle.
D, E, F are the mid points of the sides BC, CA and AB respectively of a triangle ABC.
Hence, from the mid-point theorem, the sides of ΔDEF are half the sides of Δ ABC and A( ΔDEF) is one-fourth the Δ(AABC)
D, E, F are the mid points of the sides BC, CA and AB respectively of a triangle ABC.
Hence, from the mid-point theorem, the sides of ΔDEF are half the sides of Δ ABC and A( ΔDEF) is one-fourth the Δ(AABC)
Hence, option 2.
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D, E, F are the mid points of the sides BC, CA and AB respectively of ...
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D, E, F are the mid points of the sides BC, CA and AB respectively of ...
Given:
D, E, F are the midpoints of the sides BC, CA, and AB respectively of a triangle ABC.
To find:
The ratio of the circumradii of triangles DEF and ABC.
Solution:
We know that the circumradius of a triangle is the radius of the circumcircle (the circle passing through all the vertices of the triangle).
Properties of Midpoints:
1. The line segment joining any vertex of a triangle to the midpoint of its opposite side is called a median.
2. The medians of a triangle intersect at a point called the centroid. The centroid divides each median into two segments, with the ratio of 2:1.
Centroid:
Let G be the centroid of triangle ABC. The medians AD, BE, and CF intersect at G.
Ratio of Medians:
Let's consider the ratio of the medians AG and GD. Since G divides AD in the ratio of 2:1, we can say that AG:GD = 2:1.
Similarly, the ratios of the medians BG:GE and CG:GF are also 2:1.
Ratio of Circumradii:
The ratio of the circumradii of triangles DEF and ABC can be found using the ratio of corresponding sides.
In triangle ABC, the medians AG, BG, and CG are perpendicular to the sides BC, CA, and AB respectively. Therefore, the circumradii of triangles ABC and DEF are equal to the distances AG, BG, and CG.
Since the ratios of the medians AG:GD, BG:GE, and CG:GF are all 2:1, we can say that the ratios of the circumradii AG:GD, BG:GE, and CG:GF are also 2:1.
Therefore, the ratio of the circumradii of triangles DEF and ABC is 1:2.
Answer:
The correct ratio of the circumradii of triangles DEF and ABC is 1:2, which corresponds to option B.
D, E, F are the midpoints of the sides BC, CA, and AB respectively of a triangle ABC.
To find:
The ratio of the circumradii of triangles DEF and ABC.
Solution:
We know that the circumradius of a triangle is the radius of the circumcircle (the circle passing through all the vertices of the triangle).
Properties of Midpoints:
1. The line segment joining any vertex of a triangle to the midpoint of its opposite side is called a median.
2. The medians of a triangle intersect at a point called the centroid. The centroid divides each median into two segments, with the ratio of 2:1.
Centroid:
Let G be the centroid of triangle ABC. The medians AD, BE, and CF intersect at G.
Ratio of Medians:
Let's consider the ratio of the medians AG and GD. Since G divides AD in the ratio of 2:1, we can say that AG:GD = 2:1.
Similarly, the ratios of the medians BG:GE and CG:GF are also 2:1.
Ratio of Circumradii:
The ratio of the circumradii of triangles DEF and ABC can be found using the ratio of corresponding sides.
In triangle ABC, the medians AG, BG, and CG are perpendicular to the sides BC, CA, and AB respectively. Therefore, the circumradii of triangles ABC and DEF are equal to the distances AG, BG, and CG.
Since the ratios of the medians AG:GD, BG:GE, and CG:GF are all 2:1, we can say that the ratios of the circumradii AG:GD, BG:GE, and CG:GF are also 2:1.
Therefore, the ratio of the circumradii of triangles DEF and ABC is 1:2.
Answer:
The correct ratio of the circumradii of triangles DEF and ABC is 1:2, which corresponds to option B.
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D, E, F are the mid points of the sides BC, CA and AB respectively of a triangle ABC. What is the ratio of the circumradii of triangles DEF and ABC respectively?a)1 : 4b)1 : 2c)1 : 8d)1 : 16e)2 : 3Correct answer is option 'B'. Can you explain this answer?
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