the midpoints of the sides of a triangle are (1,1),(4,3),and (3,5). find the area of the triangle. what is the answer and can u explain how u got it step by step? please...
The triangle formed by connecting the midpoints is similar to the original triangle but with all lengths half the original. Therefore the area of the triangle is four times the area of the little one inside (1/2) b h = 4 (1/2)(b/2)(h/2)
check for typos. I bet the original was supposed to be a right triangle.
To find the area of the triangle using the given midpoints of its sides, we can follow these steps:
Step 1: Determine the coordinates of the vertices.
To do this, we need to find the original coordinates of the triangle's vertices using the midpoints provided.
Let's assume the midpoints are labeled as A (1,1), B (4,3), and C (3,5).
To find the original coordinates, we can use the fact that the midpoint of a line segment is the average of its endpoints.
For example, the original coordinate of point A can be found by doubling the x-coordinate and doubling the y-coordinate of point A's midpoint:
Original coordinate of A = (2 * midpoint x-coordinate, 2 * midpoint y-coordinate)
= (2 * 1, 2 * 1)
= (2, 2)
Similarly, we can find the original coordinates of B and C:
Original coordinate of B = (2 * 4, 2 * 3)
= (8, 6)
Original coordinate of C = (2 * 3, 2 * 5)
= (6, 10)
Our triangle's vertices are A(2, 2), B(8, 6), and C(6, 10).
Step 2: Calculate the length of one side of the triangle.
We can choose any side to calculate its length. Let's calculate the length of side AB using the distance formula:
Length of AB = √((x₂ - x₁)² + (y₂ - y₁)²)
Length of AB = √((8 - 2)² + (6 - 2)²)
= √(6² + 4²)
= √(36 + 16)
= √52
= 2√13
Step 3: Calculate the altitude.
The altitude of a triangle is the perpendicular distance between a vertex and the opposite side. We can find the altitude by calculating the distance between point C and line AB.
Given points C(6, 10), and line AB has endpoints A(2, 2) and B(8, 6).
We can find the equation of line AB and then find the perpendicular line from point C.
Slope of AB = (y₂ - y₁) / (x₂ - x₁)
= (6 - 2) / (8 - 2)
= 4 / 6
= 2 / 3
The equation of line AB: y - 2 = (2 / 3)(x - 2)
The slope of a line perpendicular to AB is the negative reciprocal of 2/3, which is -3/2.
Thus, the equation of the perpendicular line from point C: y - 10 = -3/2(x - 6)
Simplifying, we get: y = -3/2x + 19
Now we have two lines, AB and the perpendicular line from point C. We can find their point of intersection to get the altitude.
Solving the equations of the two lines simultaneously, we get:
-3/2x + 19 = (2/3)x + 4
(2/3)x + (3/2)x = 19 - 4
(4/6)x + (9/6)x = 15 / (6/1)
(13/6)x = 15 / (6/1)
13x = 15
x = 15 / 13
x ≈ 1.15
Plugging the value of x back into the equation of the perpendicular line, we can find the corresponding y-coordinate:
y = -3/2(15/13) + 19
y = -45/26 + 494/26
y = 449/26
y ≈ 17.27
Therefore, the altitude from point C to side AB is ≈ 17.27 units.
Step 4: Calculate the area of the triangle.
The area of a triangle can be found using the formula:
Area = (1/2) * base * height
We can take any side of the triangle as the base and the corresponding altitude as the height. Let's take AB as the base since we have already calculated its length.
Using the length of AB ≈ 2√13 and the altitude from C ≈ 17.27:
Area = (1/2) * AB * height
= (1/2) * 2√13 * 17.27
= √13 * 17.27
≈ 31.32 square units
Thus, the area of the triangle is approximately 31.32 square units.