When a draftsman draws three lines that are to intersect at one point, the lines may not intersect as intended and subsequently will form an error triangle. If this error triangle is long and thin, one estimate for the location of the desired point is the midpoint of the shortest side. The figure shows one such error triangle.
Graph points are (3.0,1.7);(3.1,1.9);(1.8,1.5)
*A)Find an estimate for the desired intersection point.
*B) Find the length of the median for the midpoint found in part (A)
so, what is the shortest side?
Then find its midpoint.
Just use the distance formula, and recall that the midpoint's coordinates are the average of the endpoints' coordinates.
To find the estimate for the desired intersection point, we can follow these steps:
A) Find the midpoint of the shortest side:
- First, we need to determine which side of the triangle is the shortest. To do this, we can find the lengths of all three sides.
- The lengths of the three sides can be found using the distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the two points that form a side.
- Using the given coordinates, we can find the lengths of the sides:
- Side 1: d1 = sqrt((3.1-3.0)^2 + (1.9-1.7)^2) ≈ 0.227
- Side 2: d2 = sqrt((3.1-1.8)^2 + (1.9-1.5)^2) ≈ 1.459
- Side 3: d3 = sqrt((1.8-3.0)^2 + (1.5-1.7)^2) ≈ 1.408
- From the calculations, we can see that Side 1 is the shortest, with a length of approximately 0.227.
- To find the midpoint of Side 1, we can use the midpoint formula: ((x1+x2)/2, (y1+y2)/2), where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the side.
- The endpoints of Side 1 are (3.0, 1.7) and (3.1, 1.9), so applying the midpoint formula, we get:
- Midpoint of Side 1 = ((3.0+3.1)/2, (1.7+1.9)/2) = (3.05, 1.8)
- Therefore, an estimate for the desired intersection point is (3.05, 1.8).
Now, let's find the length of the median for the midpoint found in part (A).
B) Finding the length of the median:
- A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side.
- In this case, the midpoint of Side 1 (3.05, 1.8) is connected to the opposite vertex (1.8, 1.5) to form the median.
- We can calculate the length of this median using the distance formula:
- Length of the median = sqrt((3.05-1.8)^2 + (1.8-1.5)^2) ≈ 1.22
Therefore, the length of the median for the midpoint found in part (A) is approximately 1.22 units.
A) To find an estimate for the desired intersection point, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two points (x₁, y₁) and (x₂, y₂) are given by:
Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
In this case, we will use the coordinates of two points to find the midpoint. Let's choose the coordinates (3.0, 1.7) and (3.1, 1.9):
x₁ = 3.0, y₁ = 1.7
x₂ = 3.1, y₂ = 1.9
Plugging in these values into the midpoint formula:
Midpoint = ((3.0 + 3.1) / 2, (1.7 + 1.9) / 2)
= (6.1 / 2, 3.6 / 2)
= (3.05, 1.8)
Therefore, an estimate for the desired intersection point is approximately (3.05, 1.8).
B) To find the length of the median for the midpoint found in part (A), we need to find the distances between the midpoint and each of the vertices of the triangle.
Let's calculate the distances:
Distance between the midpoint and (3.0, 1.7):
d₁ = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
= sqrt((3.05 - 3.0)² + (1.8 - 1.7)²)
≈ sqrt(0.0025 + 0.01)
≈ sqrt(0.0125)
≈ 0.1118 (rounded to four decimal places)
Distance between the midpoint and (3.1, 1.9):
d₂ = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
= sqrt((3.05 - 3.1)² + (1.8 - 1.9)²)
≈ sqrt(0.0025 + 0.01)
≈ sqrt(0.0125)
≈ 0.1118 (rounded to four decimal places)
Distance between the midpoint and (1.8, 1.5):
d₃ = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
= sqrt((3.05 - 1.8)² + (1.8 - 1.5)²)
≈ sqrt(2.1025 + 0.0025)
≈ sqrt(2.105)
≈ 1.4504 (rounded to four decimal places)
Now, to find the length of the median, we take the average of the three distances:
Length of the median = (d₁ + d₂ + d₃) / 3
= (0.1118 + 0.1118 + 1.4504) / 3
= 0.558 (rounded to three decimal places)
Therefore, the length of the median for the midpoint found in part (A) is approximately 0.558.