IDZ Ryabushko 3.2 Option 6

Download Mirror

No. 1 To solve the problem, we need a formula for finding the distance between two points on a plane:

d = √((x₂-x₁)² + (y₂-y₁)²)

a) To find the equation of side AB, find the coordinates of points A and B:

A(-2, -3); B(1, 6)

Let's find the distance between points A and B:

d = √((1-(-2))² + (6-(-3))²) = √(3² + 9²) = √90

The equation for side AB is:

(x₁ - x₂)y + (y₂ - y₁)x + (x₁y₂ - x₂y₁) = 0

Substitute the coordinates of points A and B:

(-3 - 6)y + (1 - (-2))x + (-3×6 - 1×(-2)) = 0

-9y + 3x + 20 = 0

b) To find the equation for the height of CH, we find the coordinates of point H, which is the intersection of the height of CH and side AB. To do this, find the equation of line AB and the coordinates of point H:

(-3 - 6)y + (1 - (-2))x + (-3×6 - 1×(-2)) = 0

-9y + 3x + 20 = 0

The equation of line AB is:

y = (3/9)x + (20/9)

Since the height CH passes through point C and is perpendicular to side AB, the slope of the height is -3/9 = -1/3. Using the coordinates of point C and the angular coefficient of the height, we find the equation for the height of CH:

y - 1 = (-1/3)(x - 6)

y = (-1/3)x + 7

c) To find the equation of the median AM, let’s find the coordinates of point M, which is the midpoint of side AB. To do this, we find the arithmetic mean of the coordinates of points A and B:

x = (-2 + 1)/2 = -0.5; y = (-3 + 6)/2 = 1.5

Point M has coordinates (-0.5, 1.5). The equation of the median AM passes through points A and M, so we use the formula to find the equation of a straight line passing through two given points:

y - (-3) = ((1.5 - (-3))/(-0.5 - (-2)))(x - (-2))

y + 3 = (4.5/1.5)(x + 2)

y = 3x - 9

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

y = (-1/3)x + 7

y = 3x - 9

Solving the system, we obtain the coordinates of point N:

x = 1; y = 4

Point N has coordinates (1, 4).

e) To find the equation of a straight line passing through vertex C and parallel to side AB, we find the slope of side AB:

k = (6 - (-3))/(1 - (-2)) = 3

Since the desired line passes through point C, the equation of the line has the form:

y - 1 = 3(x - 6)

y = 3x - 17

f) To find the distance from point C to line AB, we use the formula:

d = |ax₀ + by₀ + c|/√(a² + b²), where x₀ and y₀ are the coordinates of point C; a, b and c are the coefficients of the equation of the line AB.

Substitute the values into the formula:

d = |(-9)×(-1) + 3×(-3) + 20|/√(9 + 81) = 6/√90 = 2√10/3.

Thus, the distance from point C to line AB is 2√10/3.

No. 2 To prove that the quadrilateral ABCD is a trapezoid, it is necessary to check that the angles between parallel sides are equal (this property is called the property of the bases of a trapezoid), that is, ∠A = ∠C and ∠B = ∠D.

Let's find the equations of sides AD and BC using the formula for finding the distance between two points on a plane:

AD: d = √((x₂-x₁)² + (y₂-y₁)²) = √((-5-3)² + (5-6)²) = √74

BC: d = √((x₂-x₁)² + (y₂-y₁)²) = √((5-(-1))² + (2-(-3))²) = √65

Since the sides AD and BC are not equal, the quadrilateral ABCD is not an isosceles trapezoid.

It remains to check that the angles between parallel sides are equal:

∠А = arctg((5-6)/(-5-3)) ≈ 128.66°

∠С = arctg((2-(-3))/(5-(-1))) ≈ 128.66°

∠В = arctg((5-6)/(3-(-1))) ≈ -41.19°

∠D = arctg((2-(-3))/(5-3)) ≈ -41.19°

Thus, ∠A = ∠C and ∠B = ∠D, which confirms that ABCD is a trapezoid.

IDZ Ryabushko 3.2 Option 6 is a digital product intended for schoolchildren and students who are preparing for exams or testing in mathematics. This product contains a selection of problems designed by experienced teachers to help students consolidate their knowledge and prepare for the exam.

Beautiful html design of the product allows you to conveniently and quickly find the information you need. All information about tasks is presented in a convenient format that allows you to quickly navigate and find the necessary data. The product also contains detailed step-by-step solutions to problems, which will help students better understand the material and find errors in their solutions.

IDZ Ryabushko 3.2 Option 6 is an ideal choice for those who want to effectively prepare for the mathematics exam and get high results. Thanks to convenient html design and high-quality content, students will be able to quickly and easily learn the material and successfully pass the exam.

***

Red Dead Redemption 2 is an exciting game from Rockstar that will allow you to immerse yourself in the cruel and dangerous world of the Wild West. Players will become part of a gang of thugs and criminals who roam America, evading the law and committing robberies.

The game has stunning graphics and sound that allow you to completely immerse yourself in the atmosphere of the Wild West. In addition, the game features various quests and missions that allow you to reveal a deep plot and learn more about the characters and the world of the game.

Red Dead Redemption 2 is region-free, meaning it can be played on any console without region restrictions. Players can enjoy the game on their console without any hindrance. This game is an excellent choice for fans of adventure games and fans of the Wild West.

***

An excellent solution for quickly and efficiently completing homework.
IDZ Ryabushko 3.2 Option 6 contains many interesting tasks that will help consolidate knowledge.
Thanks to this digital product, you can significantly reduce the time spent preparing for exams.
Option 6 is ideal for those who want to test their knowledge and skills in mathematics.
This digital product is perfect for those who prefer to learn on their own.
IDZ Ryabushko 3.2 Option 6 is an excellent choice for those who want to improve their level of knowledge in mathematics.
This digital product is very easy to use and will help you complete your homework quickly and easily.