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A point taken on each median of a triangle divides the median in the ratio 1 : 3, reckoning from the vertex.
Then the ratio of the area of the triangle with vertices at these points to that of the original triangle is
Then the ratio of the area of the triangle with vertices at these points to that of the original triangle is
- a)5 : 13
- b)25 : 64
- c)13 : 32
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
A point taken on each median of a triangle dividesthe median in the ra...
Let A (0,0); B (4m , 0) and C(4p , 4q)
M1 (2m + 2p, 2q)
M2 (2p , 2q) and M3 (2m , 0)
Let E , F and G be the point on the median.
E = (2m + 2p) / 4, 2q / 4) = ((m + p) / 2, q / 2)
F = ((2p + 12m) / 4, (2q + 0) / 4) = ((p + 6m) / 2, q / 2)
G = ((2m + 12p) / 4, (0 + 12 q) / 4) = ((m + 6p ) / 2, 3q)
Area of traingle ABC = 1/2
Area of traingle ABC = 1/2 {(0,0,0) (4m,0,1) (4p,4q,1)}
=1/2(16) = 8 unit
Area of triangle EFG = 1/2 {((m + p) / 2, q / 2, 1)) ((p + 6m) / 2, q / 2, 1) ((m + 6p)/2, 3q, 1)}
= 25/8 unit
ar (EFG) /ar (ABC) = {25 / 8} / 8
= 25/64
M1 (2m + 2p, 2q)
M2 (2p , 2q) and M3 (2m , 0)
Let E , F and G be the point on the median.
E = (2m + 2p) / 4, 2q / 4) = ((m + p) / 2, q / 2)
F = ((2p + 12m) / 4, (2q + 0) / 4) = ((p + 6m) / 2, q / 2)
G = ((2m + 12p) / 4, (0 + 12 q) / 4) = ((m + 6p ) / 2, 3q)
Area of traingle ABC = 1/2
Area of traingle ABC = 1/2 {(0,0,0) (4m,0,1) (4p,4q,1)}
=1/2(16) = 8 unit
Area of triangle EFG = 1/2 {((m + p) / 2, q / 2, 1)) ((p + 6m) / 2, q / 2, 1) ((m + 6p)/2, 3q, 1)}
= 25/8 unit
ar (EFG) /ar (ABC) = {25 / 8} / 8
= 25/64
Most Upvoted Answer
A point taken on each median of a triangle dividesthe median in the ra...
Solution:
Let ABC be the triangle with medians AD, BE, and CF, and let P, Q, and R be the points where the medians are divided in the ratio 1:3, as shown in the figure.
[Note: In what follows, we will use the fact that the medians of a triangle divide each other in the ratio 2:1. This can be proved by using the law of cosines to show that the square of the length of a median is equal to the sum of the squares of half the lengths of the other two sides of the triangle.]
1. Finding the coordinates of the points P, Q, and R:
Let A = (0,0), B = (b,0), and C = (c,d) be the vertices of the triangle, where b and c are positive numbers and d is a positive or negative number. Then the equations of the medians are:
AD: x = b/2
BE: y = d/3
CF: y = (d/2) - (c/4)
Let P = (p,q) be the point on AD such that AP/PD = 1/3. Then we have:
p = (2/3)(b/2) = b/3
q = (1/3)(0) + (2/3)(d/2) = d/3
Similarly, we can find the coordinates of Q and R:
Q = ((2/3)b, (1/3)d)
R = ((1/3)c, (2/3)(d/2)) = ((1/3)c, d/3)
2. Finding the area of the triangle with vertices P, Q, and R:
Let T be the triangle with vertices P, Q, and R. Then the side lengths of T are:
|PQ| = (1/3)b
|QR| = (1/3)sqrt(13)(d/2)
|RP| = (1/3)sqrt(13)(c/2)
To find the area of T, we can use Heron's formula:
s = (1/2)(|PQ| + |QR| + |RP|) = (1/3)(b/2 + sqrt(13)(d/4) + sqrt(13)(c/4))
A = sqrt(s(s-|PQ|)(s-|QR|)(s-|RP|))
After simplifying, we get:
A = (sqrt(3)/36)(b*sqrt(13)d - c*sqrt(13)d + 2bc)
3. Finding the ratio of the areas of T and ABC:
To find the ratio of the areas of T and ABC, we just need to divide the area of T by the area of ABC. Let M be the midpoint of BC, and let h be the altitude of ABC from A. Then we have:
Area of ABC = (1/2)bh
Area of T/ Area of ABC = (A/T)/(bh)
After substituting the expressions for A and T, we get:
Area of T/ Area of ABC = (sqrt(3)/36)(b*sqrt(13)d - c*sqrt(13)d + 2bc)/(bh)
Using the fact that AM = (2/3)h, we can eliminate d from the
Let ABC be the triangle with medians AD, BE, and CF, and let P, Q, and R be the points where the medians are divided in the ratio 1:3, as shown in the figure.
[Note: In what follows, we will use the fact that the medians of a triangle divide each other in the ratio 2:1. This can be proved by using the law of cosines to show that the square of the length of a median is equal to the sum of the squares of half the lengths of the other two sides of the triangle.]
1. Finding the coordinates of the points P, Q, and R:
Let A = (0,0), B = (b,0), and C = (c,d) be the vertices of the triangle, where b and c are positive numbers and d is a positive or negative number. Then the equations of the medians are:
AD: x = b/2
BE: y = d/3
CF: y = (d/2) - (c/4)
Let P = (p,q) be the point on AD such that AP/PD = 1/3. Then we have:
p = (2/3)(b/2) = b/3
q = (1/3)(0) + (2/3)(d/2) = d/3
Similarly, we can find the coordinates of Q and R:
Q = ((2/3)b, (1/3)d)
R = ((1/3)c, (2/3)(d/2)) = ((1/3)c, d/3)
2. Finding the area of the triangle with vertices P, Q, and R:
Let T be the triangle with vertices P, Q, and R. Then the side lengths of T are:
|PQ| = (1/3)b
|QR| = (1/3)sqrt(13)(d/2)
|RP| = (1/3)sqrt(13)(c/2)
To find the area of T, we can use Heron's formula:
s = (1/2)(|PQ| + |QR| + |RP|) = (1/3)(b/2 + sqrt(13)(d/4) + sqrt(13)(c/4))
A = sqrt(s(s-|PQ|)(s-|QR|)(s-|RP|))
After simplifying, we get:
A = (sqrt(3)/36)(b*sqrt(13)d - c*sqrt(13)d + 2bc)
3. Finding the ratio of the areas of T and ABC:
To find the ratio of the areas of T and ABC, we just need to divide the area of T by the area of ABC. Let M be the midpoint of BC, and let h be the altitude of ABC from A. Then we have:
Area of ABC = (1/2)bh
Area of T/ Area of ABC = (A/T)/(bh)
After substituting the expressions for A and T, we get:
Area of T/ Area of ABC = (sqrt(3)/36)(b*sqrt(13)d - c*sqrt(13)d + 2bc)/(bh)
Using the fact that AM = (2/3)h, we can eliminate d from the
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A point taken on each median of a triangle dividesthe median in the ratio 1 : 3, reckoning from the vertex.Then the ratio of the area of the triangle with verticesat these points to that of the original triangle isa)5 : 13b)25 : 64c)13 : 32d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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A point taken on each median of a triangle dividesthe median in the ratio 1 : 3, reckoning from the vertex.Then the ratio of the area of the triangle with verticesat these points to that of the original triangle isa)5 : 13b)25 : 64c)13 : 32d)None of theseCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A point taken on each median of a triangle dividesthe median in the ratio 1 : 3, reckoning from the vertex.Then the ratio of the area of the triangle with verticesat these points to that of the original triangle isa)5 : 13b)25 : 64c)13 : 32d)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A point taken on each median of a triangle dividesthe median in the ratio 1 : 3, reckoning from the vertex.Then the ratio of the area of the triangle with verticesat these points to that of the original triangle isa)5 : 13b)25 : 64c)13 : 32d)None of theseCorrect answer is option 'B'. Can you explain this answer?.
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