IDZ Ryabushko 3.2 Option 23

No. 1 Given vertices ∆АВС: А(–3,–1); B(–4;–5); C(8;1). Find: a) Equation of side AB; b) Equation of CH height; c) Equation of the median AM; d) Point N of intersection of the median AM and height CH; e) Equation of a line passing through vertex C and parallel to side AB; e) Distance from point C to straight line AB.

Answer:

a) The equation of side AB can be found using the coordinates of points A and B:

The equation of a line passing through two points (x1, y1) and (x2, y2) is:

y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)

For side AB:

y + 1 = (-5 + 1) / (-4 + 3) * (x + 3)

y + 1 = -4 * (x + 3)

Simplifying, we get:

y = -4x - 13

b) The equation for height CH passes through vertex C and is perpendicular to side AB. Let's find the angular coefficient of side AB:

k = (y2 - y1) / (x2 - x1) = (-5 - (-1)) / (-4 - (-3)) = -4

The angular coefficient of the height of the CH is equal to k' = -1 / k = 1 / 4.

Since the height passes through point C(8;1), its equation has the form:

y - 1 = 1 / 4 * (x - 8)

y = 1 / 4 * x - 1 / 4

c) The median AM passes through the vertex A and the middle of side BC. Let's find the coordinates of the middle of the side of the sun:

xср = (x2 + x3) / 2 = (-4 + 8) / 2 = 2

yср = (y2 + y3) / 2 = (-5 + 1) / 2 = -2

Therefore, the coordinates of point M are equal to (2;-2). The slope of the median AM is equal to:

k = (y2 - y1) / (x2 - x1) = (-2 - (-1)) / (2 - (-3)) = 1 / 5

Since the median passes through point A(–3,–1), its equation has the form:

y + 1 = 1 / 5 * (x + 3)

y = 1 / 5 * x - 4 / 5

d) The point of intersection of the median AM and the height CH is the center of gravity of the triangle and divides the median in a ratio of 2:1. Let's find the coordinates of point N:

xN = (xA + xM*2) / 3 = (-3 + 2*2) / 3 = -1/3

yN = (yA + yM*2) / 3 = (-1 + 2*(-2))/ 3 = -5 / 3

Point N has coordinates (-1/3; -5/3).

e) The equation of a straight line passing through vertex C and parallel to side AB has the same slope as the equation of side AB:

y - y1 = -4 * (x - x1)

Substitute the coordinates of point C(8;1):

y - 1 = -4 * (x - 8)

y = -4x + 33

e) The distance from point C to straight AB is equal to the distance from point C to the projection of point C onto straight AB. Let's find the coordinates of the projection of point C onto line AB:

xпр = (k^2 * xC - k * yC - k * b) / (k^2 + 1) = (-4^2 * 8 - (-4) * 1 - (-13)) / (16 + 1) = -59 / 17

ypr = k * xpr + b = -4 * (-59 / 17) - 13 = 95 / 17

The distance from point C to line AB is equal to the distance between points C and its projection onto line AB:

d = sqrt((xC - xpr)^2 + (yC - ypr)^2) = sqrt((8 + 59 / 17)^2 + (1 - 95 / 17)^2) = 17 / sqrt(170)

Answer:

a) y = -4x - 13; b) y = 1/4 * x - 1/4; c) y = 1/5 * x - 4/5; d) N(-1/3; -5/3); e) y = -4x + 33; e) d = 17 / sqrt(170). No. 2 Write down the equation of the straight line passing through the point A(–2;3) and the angle components with the Ox axis: a) 45°; b) 90°; c) 0°.

Answer:

The angle between the straight line and the Ox axis can be found using the slope k:

k = tan(α), where α is the angle between the straight line and the Ox axis

a) At α = 45°, k = 1.

The equation of a straight line passing through the point A(–2;3) and having an angular coefficient k = 1 has the form:

y - y1 = k * (x - x1)

y - 3 = 1 * (x + 2)

y = x + 5

b) At α = 90°, k = infinity.

The straight line passing through the point A(–2;3) and parallel to the Oy axis has the equation:

x = -2

c) At α = 0°, k = 0.

The straight line passing through the point A(–2;3) and parallel to the Ox axis has the equation:

y = 3

Answer:

a) y = x + 5; b) x = -2

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IDZ Ryabushko 3.2 Option 23 is a set of problems in mathematics, which includes the following tasks:

1. The vertices of the triangle ∆ABC are given: A(–3,–1); B(–4;–5); C(8;1). Necessary: a) Find the equation of side AB. b) Find the equation for the height of the CH. c) Find the equation of the median AM. d) Find the point N of the intersection of the median AM and the height CH. e) Find the equation of the line passing through vertex C and parallel to side AB. f) Find the distance from point C to line AB.

2. It is necessary to write down the equation of the straight line passing through the point A(–2;3) and making an angle with the Ox axis: a) 45°; b) 90°; c) 0°.

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