To solve the problem of finding the number of obtuse triangles that can be formed with the points (8,9), (8,16), and (20,25) as the feet of perpendiculars drawn from the vertices on the opposite sides, we can follow these steps:
### Step 1: Identify the Points
The given points are:
- A(8, 9)
- B(8, 16)
- C(20, 25)
### Step 2: Calculate the Distances Between Points
We need to calculate the distances between the points A, B, and C to understand the triangle formed by these feet of perpendiculars.
1. **Distance AB**:
\[
AB = \sqrt{(8 - 8)^2 + (16 - 9)^2} = \sqrt{0 + 7^2} = 7
\]
2. **Distance BC**:
\[
BC = \sqrt{(20 - 8)^2 + (25 - 16)^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15
\]
3. **Distance CA**:
\[
CA = \sqrt{(20 - 8)^2 + (25 - 9)^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20
\]
### Step 3: Determine the Type of Triangle
To determine if the triangle is obtuse, we can use the property that in a triangle with sides \( a, b, c \) (where \( c \) is the longest side), the triangle is obtuse if:
\[
c^2 > a^2 + b^2
\]
Let’s assign:
- \( a = AB = 7 \)
- \( b = BC = 15 \)
- \( c = CA = 20 \)
Now, we check:
\[
c^2 = 20^2 = 400
\]
\[
a^2 + b^2 = 7^2 + 15^2 = 49 + 225 = 274
\]
Since \( 400 > 274 \), the triangle formed by points A, B, and C is obtuse.
### Step 4: Count the Number of Obtuse Triangles
The problem states that we need to find the number of obtuse triangles that can be formed with these points as feet of perpendiculars.
For any triangle formed by three points, there are three possible obtuse triangles depending on which vertex is opposite the longest side. In our case, since we have only one triangle formed by these three points, we can conclude that:
- There is **only one obtuse triangle** that can be formed with the given points.
### Final Answer
The number of obtuse triangles that can be formed is **1**.
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