1.The midpoints of the sides of a triangle are (5,1), (3,-5) and (-5, -1).
Find the coordinates of the Vertices of the triangle.
please give me the answer.
Answers
Answer:
The coordinates of the sides of the triangle are A (-3, 5), B (13, -3), and C (5, -7)
Step-by-step explanation:
Given: The midpoints of the sides of a triangle are (5,1), (3,-5), and (-5, -1).
Let the midpoint of AB, BC, and AC be (5,1), (3,-5), and (-5, -1) respectively.
To Calculate: The coordinates of the Vertices of the triangle.
Solution:
Let the coordinates of the vertices of the triangles A, B and C be
and
So, Using the midpoint formula, we get
equation (1)
And
equation (2)
Similarly, we have :
equation (3)
equation (4)
And,
equation (5)
equation (6)
Now,
Adding equations 1, 3 and 5
And Adding equations 2, 4 and 6
We get,
equation (7)
And,
equation (8)
Now, On Solving these equations, we get
On, (7) - (1)
On, (7) - (3)
On, (7) - (5)
Now,
On, (8) - (2)
On, (8) - (4)
On, (8) - (6)
Thus, the coordinates of the sides of the triangle are A (-3, 5), B (13, -3), and C (5, -7)
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Answer:
The vertex of ABC triangle is A=(7,6), B=(23,-2), C=(3,-6).
Step-by-step explanation:
As per question we need to find the vertex of the triangle.
Let the triangle is ABC. A,B,C is the vertex of the triangle.
The given data is midpoints of the sides of a triangle are (5,1), (3,-5) and (-5, -1).
Let the point (5,1) is the middle point of the side AB, and the point (3,-5) is the middle point of the side BC, and the point (-5,-1).
Now also let the coordinates of the vertex is A=(x₁,y₁), B=(x₂,y₂), C=(x₃,y₃).
we know that the middle point of the side AB is
and the middle point of BC is and the middle point of AC is .
now from AB, BC, CA sides we get
x₁ +x₂ =10.......... (1),
y₁+y₂ =2............ (2),
x₂ +x₃ =6.........(3),
y₂ +y₃ = -10........(4),
x₃+x₁=-10..........(5),
y₃ +y₁= -2...........(6)
add equation (1), (3), and (5) we get
2(x₁ +x₂ + x₃) =26
or, (x₁ +x₂ + x₃)=13......(7)
now put the x₁ +x₂ =10 in equation (7) we get
x₃=3,
now put the x₂ +x₃ =6 in equation (7) we get
x₁=7
now put the x₃+x₁=-10 in equation (7) we get
x₂=23
continuing this process we get
y₁+y₂+y₃=-4......(8)
from (8) we get
y₁=6
y₂=-2
y₃ =-6
Therefore the vertex of ABC triangle is A=(7,6), B=(23,-2), C=(3,-6).
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