the midpoints of the sides of a triangle are (1,1),(4,3),and (3,5). find the area of the triangle. please answer and show how u did it step by step please!!!! how do i find area of the smaller triangle?
see your previous post of this
To find the area of a triangle using the midpoints of its sides, you can follow these steps:
Step 1: Find the coordinates of the vertices of the original triangle.
Let the midpoints of the sides be A(1,1), B(4,3), and C(3,5). To find the coordinates of the vertices, you need to double the x and y coordinates of the midpoints.
The vertices are:
A' (2,2) - double the x and y coordinates of A.
B' (8,6) - double the x and y coordinates of B.
C' (6,10) - double the x and y coordinates of C.
Step 2: Find the lengths of the sides of the triangle.
Using the distance formula, you can find the lengths of the sides AB, BC, and AC. The distance formula is given by:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
AB' = √((8 - 2)^2 + (6 - 2)^2) = √(6^2 + 4^2) = √(36 + 16) = √52
BC' = √((6 - 8)^2 + (10 - 6)^2) = √((-2)^2 + 4^2) = √(4 + 16) = √20
AC' = √((6 - 2)^2 + (10 - 2)^2) = √(4^2 + 8^2) = √(16 + 64) = √80
Step 3: Use Heron's formula to find the area of the triangle.
Heron's formula states that the area of a triangle with side lengths a, b, and c can be calculated using the formula:
Area = √(s(s - a)(s - b)(s - c))
where s is the semi-perimeter of the triangle, given by:
s = (a + b + c) / 2
In the case of our triangle, the side lengths are AB', BC', and AC', so:
a = AB' = √52
b = BC' = √20
c = AC' = √80
The semi-perimeter, s, is calculated as:
s = (a + b + c) / 2 = (√52 + √20 + √80) / 2
Now, substitute these values into Heron's formula:
Area = √(s(s - a)(s - b)(s - c))
Step 4: Calculate the area of the smaller triangle (ABC).
Using the formula from Step 3, substitute the values of a, b, c, and s:
Area = √(s(s - a)(s - b)(s - c))
Now, you can plug in the values and evaluate to find the area of the triangle.