4.1 Q-17

Question Statement

Find the incenter of the triangle formed by the points $A(4, -2)$, $B(-2, 4)$, and $C(5, 5)$. Also, determine the center of the triangle.

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Background and Explanation

In this problem, we are asked to find the incenter of the triangle, which is the point of intersection of the angle bisectors of the triangle. The incenter is equidistant from all sides of the triangle. To solve this, we will use the concept of weighted averages to find the coordinates of the incenter. The weights are the lengths of the sides opposite to each vertex of the triangle.

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Solution

We are given the coordinates of the vertices of the triangle:

• $A(4, -2)$
• $B(-2, 4)$
• $C(5, 5)$

To find the incenter, we need to:

1. Find the lengths of the sides of the triangle.

   We can use the distance formula to calculate the lengths of the sides:

   • The length of side $AB$:

$$AB = \sqrt{(-2 - 4)^2 + (4 + 2)^2} = \sqrt{(-6)^2 + (6)^2} = \sqrt{36 + 36} = 6\sqrt{2}$$

• The length of side $BC$:

$$BC = \sqrt{(5 + 2)^2 + (5 - 4)^2} = \sqrt{(7)^2 + (1)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2}$$

• The length of side $AC$:

$$AC = \sqrt{(5 + 2)^2 + (5 + 2)^2} = \sqrt{(7)^2 + (7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}$$

2. Use the formula for the incenter.

   The coordinates of the incenter $I(x, y)$ are given by the weighted averages of the coordinates of the vertices, with the side lengths as weights:

$$x = \frac{ax_1 + bx_2 + cx_3}{a + b + c}$$ $$y = \frac{ay_1 + by_2 + cy_3}{a + b + c}$$

where $a = BC = 5\sqrt{2}$, $b = AC = 7\sqrt{2}$, and $c = AB = 6\sqrt{2}$.

Substituting the values into the formulas for $x$ and $y$:

• For $x$:

$$x = \frac{(5\sqrt{2})(4) + (7\sqrt{2})(-2) + (6\sqrt{2})(5)}{5\sqrt{2} + 7\sqrt{2} + 6\sqrt{2}} = \frac{20\sqrt{2} - 14\sqrt{2} + 30\sqrt{2}}{18\sqrt{2}} = \frac{36\sqrt{2}}{18\sqrt{2}} = 2$$

• For $y$:

$$y = \frac{(5\sqrt{2})(-2) + (7\sqrt{2})(4) + (6\sqrt{2})(5)}{5\sqrt{2} + 7\sqrt{2} + 6\sqrt{2}} = \frac{-10\sqrt{2} + 28\sqrt{2} + 30\sqrt{2}}{18\sqrt{2}} = \frac{48\sqrt{2}}{18\sqrt{2}} = \frac{8}{3}$$

Thus, the coordinates of the incenter are $\left( 2, \frac{8}{3} \right)$.

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Key Formulas or Methods Used

• Distance Formula to find the lengths of the sides of the triangle:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

• Incenter Formula to find the coordinates of the incenter:

$$x = \frac{ax_1 + bx_2 + cx_3}{a + b + c}, \quad y = \frac{ay_1 + by_2 + cy_3}{a + b + c}$$

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Summary of Steps

1. Calculate the lengths of the sides $AB$, $BC$, and $AC$ using the distance formula.
2. Use the formula for the incenter to find the coordinates of the incenter using the side lengths as weights.
3. The coordinates of the incenter are $\left( 2, \frac{8}{3} \right)$.

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