To find the x-coordinate of the incenter of the triangle whose midpoints of the sides are given as (0,1), (1,1), and (1,0), we can follow these steps:
### Step 1: Identify the vertices of the triangle
The midpoints of the sides of the triangle are given. Let's denote these midpoints as:
- A = (0, 1)
- B = (1, 1)
- C = (1, 0)
To find the vertices of the triangle, we can use the fact that the midpoint of a segment is the average of the coordinates of the endpoints.
### Step 2: Find the vertices
1. The midpoint A (0, 1) is the midpoint of the side connecting vertices (x1, y1) and (x2, y2).
2. The midpoint B (1, 1) is the midpoint of another side connecting vertices (x3, y3) and (x4, y4).
3. The midpoint C (1, 0) is the midpoint of the last side connecting vertices (x5, y5) and (x6, y6).
By plotting these points, we can deduce the vertices of the triangle:
- The vertices can be determined as follows:
- The vertex opposite to midpoint A (0, 1) lies on the line connecting (1, 1) and (1, 0). Thus, we can deduce that the vertices are (0, 0), (2, 0), and (0, 2).
### Step 3: Calculate the lengths of the sides
Now, we can calculate the lengths of the sides of the triangle:
- Length of side opposite to vertex (0, 0):
- Distance between (2, 0) and (0, 2) = √[(2-0)² + (0-2)²] = √[4 + 4] = √8 = 2√2
- Length of side opposite to vertex (2, 0):
- Distance between (0, 0) and (0, 2) = √[(0-0)² + (2-0)²] = √[4] = 2
- Length of side opposite to vertex (0, 2):
- Distance between (0, 0) and (2, 0) = √[(2-0)² + (0-0)²] = √[4] = 2
### Step 4: Use the formula for the inradius
The inradius \( r \) of a triangle can be calculated using the formula:
\[ r = \frac{a + b - c}{2} \]
Where \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle.
Here, we have:
- \( a = 2 \)
- \( b = 2 \)
- \( c = 2\sqrt{2} \)
Substituting these values into the formula:
\[ r = \frac{2 + 2 - 2\sqrt{2}}{2} = \frac{4 - 2\sqrt{2}}{2} = 2 - \sqrt{2} \]
### Step 5: Determine the x-coordinate of the incenter
Since the incenter of a triangle is located at the coordinates (r, r) where r is the inradius, the x-coordinate of the incenter is:
\[ x = r = 2 - \sqrt{2} \]
### Final Answer
The x-coordinate of the incenter of the triangle is \( 2 - \sqrt{2} \).
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