IDZ Ryabushko 3.2 Option 28

№1

The vertices ∆АВС are given: А(1;–3); B(0;7); C(–2;4). Find:

1. Equation of side AB;
2. CH height equation;
3. AM the equation media;
4. Point N of intersection of the median AM and height CH;
5. Equation of a line passing through vertex C and parallel to side AB;
6. Distance from point C to straight line AB.

Answer:

To solve the problem we need the following mathematical formulas:

• Equation of a line passing through two given points:

(and - and₁) / (x - x₁) = (y₂ - y₁) / (x₂ - x₁)

The equation of a straight line in general form:

Ax + By + C = 0

Distance from point to line:

d = |Ax₀ + By₀ + C| / √(A² + B²)

a) Equation of side AB:

To find the equation of side AB, we need to find the coefficients of the equation of the straight line passing through points A(1;–3) and B(0;7).

Let's first find the slope of the line:

k = (and₂ - y₁) / (x₂ - x₁) = (7 - (-3)) / (0 - 1) = -10

Then we find the free coefficient:

b = y₁ - kx₁ = -3 - (-10) * 1 = 7

Thus, the equation of side AB is:

y = -10x + 7

b) CH height equation:

To find the equation for the height CH, we need to find the coefficients of the equation of the straight line passing through the point C(–2;4) and perpendicular to side AB.

Let us first find the angular coefficient of the straight line perpendicular to side AB:

k_AB = (y₂ - y₁) / (x₂ - x₁) = (7 - (-3)) / (0 - 1) = -10

The angular coefficient of a straight line perpendicular to side AB is equal to k_CD = 1 / k_AB = -1 / (-10) = 1/10.

Then we find the free coefficient:

b = y₁ - k_CDx₁ = 4 - (1/10) * (-2) = 4.2

Thus, the CH height equation has the form:

y = (1/10)x + 4.2

(c) The equation for media is:

To find the equation of the median AM, we need to find the coefficients of the equation of the straight line passing through the point M (the middle of side AB) and the vertex C(–2;4).

Let's first find the coordinates of point M:

x_M = (x_A + x_B) / 2 = (1 + 0) / 2 = 0.5

y_M = (y_A + y_B) / 2 = (-3 + 7) / 2 = 2

Thus, the coordinates of point M are equal to (0.5;2).

Let's find the slope of the line:

k = (and₂ - y₁) / (x₂ - x₁) = (4 - 2) / (-2 - 0.5) = -4/5

Then we find the free coefficient:

b = y₁ - kx₁ = 4 - (-4/5) * (-2) = 4.6

Thus, the equation of media AM has the form:

y = (-4/5)x + 4.6

d) Point N of intersection of the median AM and height CH:

To find the intersection point of the AM median and the CH height, we need to solve the system of equations:

{ y = (-4/5)x + 4.6

{ y = (1/10)x + 4.2

Solving this system of equations, we get:

x = 10, y = 8

Thus, point N has coordinates (10;8).

e) Equation of a line passing through vertex C and parallel to side AB:

To find the equation of a line passing through vertex C and parallel to side AB, we can use the slope of the line passing through points A and B:

k = (and₂ - y₁) / (x₂ - x₁) = (7 - (-3)) / (0 - 1) = -10

Then we find the free coefficient:

b = y₁ - kx₁ = 4 - (-10) * (-2) = -16

Thus, the equation of a line passing through vertex C and parallel to side AB has the form:

y = -10x - 16

f) Distance from

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