The vertices of a triangle are `(2,1), (5,2) and (4,4)` The lengths of the perpendicular from these vertices on the opposite sides are

Options

A. `(7)/(sqrt5),(7)/(sqrt13),(7)/(sqrt6)`
B. `(7)/(sqrt6),(7)/(sqrt8),(7)/(sqrt10)`
C. `(7)/(sqrt5),(7)/(sqrt8),(7)/(sqrt15)`
D. `(7)/(sqrt5),(7)/(sqrt13),(7)/(sqrt10)`

Answer

To solve the problem of finding the lengths of the perpendiculars from the vertices of the triangle with vertices A(2, 1), B(5, 2), and C(4, 4) to the opposite sides, we will follow these steps:

### Step 1: Find the equations of the sides of the triangle.

1. **Equation of side BC:**
- Points B(5, 2) and C(4, 4).
- Slope (m) = (y2 - y1) / (x2 - x1) = (4 - 2) / (4 - 5) = 2 / -1 = -2.
- Using point-slope form: y - y1 = m(x - x1).
- Using point B(5, 2): y - 2 = -2(x - 5).
- Rearranging gives: 2x + y - 12 = 0.

2. **Equation of side AC:**
- Points A(2, 1) and C(4, 4).
- Slope (m) = (4 - 1) / (4 - 2) = 3 / 2.
- Using point-slope form: y - 1 = (3/2)(x - 2).
- Rearranging gives: 3x - 2y = 1.

3. **Equation of side AB:**
- Points A(2, 1) and B(5, 2).
- Slope (m) = (2 - 1) / (5 - 2) = 1 / 3.
- Using point-slope form: y - 1 = (1/3)(x - 2).
- Rearranging gives: x - 3y + 7 = 0.

### Step 2: Calculate the perpendicular distances from each vertex to the opposite side.

1. **Distance from A(2, 1) to line BC (2x + y - 12 = 0):**
- Distance formula: \( d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \)
- Here, A = 2, B = 1, C = -12, and point (x1, y1) = (2, 1).
- \( d = \frac{|2(2) + 1(1) - 12|}{\sqrt{2^2 + 1^2}} = \frac{|4 + 1 - 12|}{\sqrt{4 + 1}} = \frac{|-7|}{\sqrt{5}} = \frac{7}{\sqrt{5}} \).

2. **Distance from B(5, 2) to line AC (3x - 2y - 1 = 0):**
- A = 3, B = -2, C = -1, and point (x1, y1) = (5, 2).
- \( d = \frac{|3(5) - 2(2) - 1|}{\sqrt{3^2 + (-2)^2}} = \frac{|15 - 4 - 1|}{\sqrt{9 + 4}} = \frac{|10|}{\sqrt{13}} = \frac{10}{\sqrt{13}} \).

3. **Distance from C(4, 4) to line AB (x - 3y + 7 = 0):**
- A = 1, B = -3, C = 7, and point (x1, y1) = (4, 4).
- \( d = \frac{|1(4) - 3(4) + 7|}{\sqrt{1^2 + (-3)^2}} = \frac{|4 - 12 + 7|}{\sqrt{1 + 9}} = \frac{| -1 |}{\sqrt{10}} = \frac{1}{\sqrt{10}} \).

### Final Results:
- Length of the perpendicular from A to BC: \( \frac{7}{\sqrt{5}} \)
- Length of the perpendicular from B to AC: \( \frac{10}{\sqrt{13}} \)
- Length of the perpendicular from C to AB: \( \frac{1}{\sqrt{10}} \)