The image appears to contain a problem in Latvian about a triangle whose vertices are defined by the coordinate origin and the intersection points of the line $y = -2x + 6$ with the $Ox$ and $Oy$ axes. Here's a step-by-step solution to the task:

1. Find the vertices of the triangle

The line $y = -2x + 6$ intersects the axes as follows:

Intersection with $Oy$-axis:

• At the $Oy$-axis, $x = 0$.
• Substituting into the equation $y = -2(0) + 6$: $y = 6$ So the intersection point is $(0, 6)$.

Intersection with $Ox$-axis:

• At the $Ox$-axis, $y = 0$.
• Solving $0 = -2x + 6$: $x = 3$ So the intersection point is $(3, 0)$.

Thus, the vertices of the triangle are:

• $A = (0, 0)$ (origin),
• $B = (3, 0)$,
• $C = (0, 6)$.

2. Calculate the area of the triangle

The formula for the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$ is: $\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

Substituting $A = (0, 0)$, $B = (3, 0)$, and $C = (0, 6)$: $\text{Area} = \frac{1}{2} \left| 0(0 - 6) + 3(6 - 0) + 0(0 - 0) \right|$ $\text{Area} = \frac{1}{2} \left| 0 + 18 + 0 \right| = \frac{1}{2} \cdot 18 = 9$

So, the area of the triangle is $9 \, \text{square units}$.

3. Write the equation of the midline parallel to the longest side

The longest side of the triangle is $BC$, which connects $B = (3, 0)$ and $C = (0, 6)$. The slope of $BC$ is: $\text{slope of } BC = \frac{6 - 0}{0 - 3} = -2$

The midline parallel to $BC$ passes through the midpoint of the opposite side $AB$, whose midpoint is: $\text{Midpoint of } AB = \left( \frac{0 + 3}{2}, \frac{0 + 0}{2} \right) = \left( \frac{3}{2}, 0 \right)$

Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, with slope $-2$ and point $\left( \frac{3}{2}, 0 \right)$: $y - 0 = -2\left(x - \frac{3}{2}\right)$ $y = -2x + 3$

Thus, the equation of the midline is: $y = -2x + 3$

Final Results:

1. Area of the triangle: $9 \, \text{square units}$.
2. Equation of the midline parallel to the longest side: $y = -2x + 3$.

Would you like further clarification or details on any of the steps? 😊

Related Questions:

1. How do you calculate the distance between two points in a coordinate plane?
2. Can you prove why the midline is always parallel to one of the triangle’s sides?
3. What are the general properties of triangles formed by intersecting a line with the coordinate axes?
4. Can you derive the equation of a line given two points on it?
5. How would the problem change if the line equation were $y = mx + c$?

Tip:

Always sketch the triangle to visualize relationships between its sides, vertices, and midlines!