No. 1. Given vertices ∆АВС: А(–4;2); B(6;–4); C(4;10). Find:
a) Equation of side AB: In order to find the equation of side AB, you need to find the coefficients of the equation of the straight line passing through points A and B. To do this, we use the formula: y - y1 = (y2 - y1) / (x2 - x1) * ( x - x1), where (x1, y1) are the coordinates of point A, and (x2, y2) are the coordinates of point B.
We have: (x1, y1) = (-4, 2), (x2, y2) = (6, -4) The equation for side AB is: y - 2 = (-4 - 2) / (6 - (-4 )) * (x - (-4)) y - 2 = -6/10 * (x + 4) y = -3/5 * x + 2
b) Equation of height CH: In order to find the equation of height CH, you need to find the coefficients of the equation of a straight line passing through point C and perpendicular to side AB. To do this, we use the property: if a straight line with the equation y = kx + b is perpendicular to the straight line with the equation y = k'x + b', then k * k' = -1.
From the equation of side AB, we know that its slope is -3/5. This means that the slope coefficient of the CH height is 5/3. Point C lies at a height, which means we know the coordinates of point C. Then the equation for the height of CH has the form: y - 10 = 5/3 * (x - 4) y = 5/3 * x - 10/3
c) Equation of the median AM: In order to find the equation of the median AM, you need to find the coordinates of the point M - the middle of the side BC and the coefficients of the equation of the straight line passing through points A and M. To do this, we use the formulas: x = (x1 + x2) / 2 , y = (y1 + y2) / 2 - coordinates of point M, and the equation of the straight line passing through points A and M: y - y1 = (y2 - y1) / (x2 - x1) * (x - x1), where ( x1, y1) are the coordinates of point A, and (x2, y2) are the coordinates of point M.
We have: (x1, y1) = (-4, 2), (x2, y2) = (5, 3) Point M has coordinates: x = (-4 + 5) / 2 = 1/2, y = (2 + 3) / 2 = 5/2 The equation for the AM median is: y - 2 = (3 - 2) / (5 - (-4)) * (x - (-4)) y - 2 = 1/9 * (x + 4) y = 1/9 * x + 22/9
d) Point N of the intersection of the median AM and the height CH: Point N is the point of intersection of the median AM and the height CH. Let's find its coordinates by solving a system of equations composed of the equations of the AM median and CH height: { y = 5/3 * x - 10/3, y = 1/9 * x + 22/9 }
Having solved the system of equations, we get: x = 18/7, y = 20/7
Point N has coordinates (18/7, 20/7).
e) Equation of a line passing through vertex C and parallel to side AB: A line passing through vertex C and parallel to side AB has the same slope coefficient as side AB (-3/5). Let's find the coefficient b of the equation of this line by substituting the coordinates of point C and the value of the slope coefficient into the equation: 10 = -3/5 * 4 + b b = 58/5
The equation of a straight line passing through vertex C and parallel to side AB is: y = -3/5 * x + 58/5
f) Distance from point C to line AB: In order to find the distance from point C to line AB, you need to find a perpendicular to side AB passing through point C and find the point of intersection of this perpendicular with side AB. Then you can find the distance between point C and the found intersection point.
Let us find the equation of a perpendicular to side AB passing through point C. The slope coefficient of this perpendicular is equal to the inverse value of the slope coefficient of side AB (5/3). Point C lies on this perpendicular, which means we know the coordinates of point C. Then the equation of the perpendicular looks like: y - 10 = -3/5 * (x - 4) y = -3/5 * x + 22/5
Let's find the point of intersection of the perpendicular and side AB by solving a system of equations composed of the equations of the perpendicular and side AB: { y = -3/5 * x + 22/5, y = -3/5 * x + 2 }
Having solved the system of equations, we get: x = 2, y = 4
The intersection point of the perpendicular and side AB has coordinates (2, 4). The distance from point C to line AB is equal to the distance from point C to the found intersection point, which can be found using the formula for the distance between two points: d = sqrt((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) are the coordinates of point C, and (x2, y2) are the coordinates of the found intersection point.
We have: (x1, y1) = (4, 10), (x2, y2) = (2, 4) d = sqrt((2 - 4)^2 + (4 - 10)^2) = sqrt(20) = 2*sqrt(5)
Answer: a) y = -3/5 * x + 2; b) y = 5/3 * x - 10/3; c) y = 1/9 *
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IDZ Ryabushko 3.2 Option 18 is a set of tasks in mathematics, which includes solving problems for finding equations of lines, heights, medians, distances, etc.
In task No. 1, the vertices of triangle ABC are given and you need to find the equations of side AB, height CH, median AM, the point of intersection of the median AM and height CH, the equation of the line passing through vertex C and parallel to side AB, as well as the distance from point C to line AB .
In task No. 2 you need to create an equation of a line passing through the origin of coordinates and the intersection point of two given lines.
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