The points (3,4), (11, 10) and (5, 11/2) are

Options

A. collinear
B. vertices of an equilateral triangle
C. vertices of isosceles triangle
D. vertices of scalene triangle.

Answer

To determine whether the points (3, 4), (11, 10), and (5, 11/2) are collinear, we can follow these steps:

### Step 1: Define the Points
Let:
- Point P = (3, 4)
- Point Q = (11, 10)
- Point R = (5, 11/2)

### Step 2: Use the Distance Formula
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 3: Calculate the Distance PQ
Using the distance formula to find the distance between points P and Q:
\[ PQ = \sqrt{(11 - 3)^2 + (10 - 4)^2} \]
Calculating this:
\[ PQ = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Step 4: Calculate the Distance QR
Now, calculate the distance between points Q and R:
\[ QR = \sqrt{(5 - 11)^2 + \left(\frac{11}{2} - 10\right)^2} \]
Calculating this:
\[ QR = \sqrt{(-6)^2 + \left(\frac{11 - 20}{2}\right)^2} = \sqrt{36 + \left(-\frac{9}{2}\right)^2} = \sqrt{36 + \frac{81}{4}} \]
Finding a common denominator:
\[ QR = \sqrt{\frac{144}{4} + \frac{81}{4}} = \sqrt{\frac{225}{4}} = \frac{15}{2} \]

### Step 5: Calculate the Distance PR
Next, calculate the distance between points P and R:
\[ PR = \sqrt{(5 - 3)^2 + \left(\frac{11}{2} - 4\right)^2} \]
Calculating this:
\[ PR = \sqrt{(2)^2 + \left(\frac{11 - 8}{2}\right)^2} = \sqrt{4 + \left(\frac{3}{2}\right)^2} = \sqrt{4 + \frac{9}{4}} \]
Finding a common denominator:
\[ PR = \sqrt{\frac{16}{4} + \frac{9}{4}} = \sqrt{\frac{25}{4}} = \frac{5}{2} \]

### Step 6: Check for Collinearity
To check if the points are collinear, we need to verify if:
\[ PQ = PR + QR \]
Substituting the values we found:
\[ 10 = \frac{5}{2} + \frac{15}{2} \]
Calculating the right side:
\[ \frac{5}{2} + \frac{15}{2} = \frac{20}{2} = 10 \]
Since both sides are equal, the points are collinear.

### Conclusion
The points (3, 4), (11, 10), and (5, 11/2) are collinear.

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