Question

How are the points (3, 5) and (5, 3) different?

Answer 1

Answer on Question #83542 – Math – Analytic Geometry

Question

How are the points (3, 5) and (5, 3) different?

Solution

Two points are said to be symmetrical with respect to a given point when the given point bisects the line joining the two points. The given point is called the center of symmetry.

Two points are said to be symmetrical with respect to a given line when the given line is the perpendicular bisector of the line joining the two points. The given line is called the axis of symmetry.

We have two points $A(3,5)$ and $B(5,3)$. Find the coordinates of the point $C$ bisecting a line segment $AB$ from $(3,5)$ to $B(5,3)$

$$x_C = \frac{x_A + x_B}{2}, \quad y_C = \frac{y_A + y_B}{2}$$ $$x_C = \frac{3 + 5}{2} = 4, \quad y_C = \frac{5 + 3}{2} = 4, \quad \text{point } C(4,4)$$

Two points $A(3,5)$ and $B(5,3)$ are symmetrical with respect to point $C(4,4)$. Find the slope of the line passing through the points $A(3,5)$ and $B(5,3)$

$$\text{slope}_1 = m_1 = \frac{y_B - y_A}{x_B - x_A} = \frac{3 - 5}{5 - 3} = -1$$

If two lines with slopes $m_1$ and $m_2$ are perpendicular, then

$$m_1 m_2 = -1$$

Find the slope of the line which is perpendicular to the line $AB$

$$\text{slope}_2 = m_2 = -\frac{1}{m_1} = -\frac{1}{-1} = 1$$

The equation of the line perpendicular to the line $AB$

$$y = x + b$$

Find the equation of the line perpendicular to the line $AB$ and passing through the point $C(4,4)$

$$4 = 4 + b \Rightarrow b = 0$$ $$y = x$$

Therefore, two points $A(3,5)$ and $B(5,3)$ are symmetrical with respect to the line $y = x$.

Find the distance $d_{AB}$

$$d_{AB} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(5 - 3)^2 + (3 - 5)^2} = 2\sqrt{2}$$

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