Question

given triangle GHI with G(4,-3), H(-4,2) and I(2,4), find the perpendicular bisector of HI in standard form.

Answer 1

3x + y = 0.

Explanation:

To find the perpendicular bisector of segment HI in standard form, we need to follow a few steps: Step 1: Find the midpoint of segment HI. - The midpoint is found by averaging the x-coordinates and the y-coordinates of the endpoints. - The x-coordinate of the midpoint is (x₁ + x₂) / 2, and the y-coordinate is (y₁ + y₂) / 2. Given the endpoints H(-4, 2) and I(2, 4), we can find the midpoint: - x-coordinate: (-4 + 2) / 2 = -2 / 2 = -1 - y-coordinate: (2 + 4) / 2 = 6 / 2 = 3 So, the midpoint of segment HI is M(-1, 3). Step 2: Find the slope of segment HI. - The slope of a line passing through two points can be found using the formula: (y₂ - y₁) / (x₂ - x₁). Given the points H(-4, 2) and I(2, 4), we can find the slope: - slope = (4 - 2) / (2 - (-4)) = 2 / 6 = 1/3 Step 3: Find the negative reciprocal of the slope. - The negative reciprocal of a slope is obtained by flipping the fraction and changing the sign. The negative reciprocal of 1/3 is -3/1, which is -3. Step 4: Use the midpoint and negative reciprocal slope to write the equation of the perpendicular bisector in standard form (Ax + By = C). - The standard form of a linear equation is Ax + By = C, where A, B, and C are constants. Using the midpoint M(-1, 3) and the slope -3, we can write the equation of the perpendicular bisector. - Substitute the coordinates of the midpoint (-1, 3) into the equation y - y₁ = m(x - x₁). - We get y - 3 = -3(x - (-1)). Expanding the equation gives: y - 3 = -3(x + 1), y - 3 = -3x - 3, 3x + y = 0 So, the equation of the perpendicular bisector of segment HI in standard form is 3x + y = 0.

Answer 2

The perpendicular bisector would be the line that connects with the line HI at a 90 degree angle and in the mid-point. This means that we would have to find the slope of the line HI and take the opposite. The slope of HI is 1/3. This means that the slope of the perpendicular line is -3. Now we must solve for the line that crosses through the midpoint. y=mx+b we know that m=-3 and that the midpoint is at -1,3 so x=-1 and y=3.  3=(-3)-1+b. We then solve for b and find it to be 0.  So the final answer is y= -3x