Question

plzz can u guys help me with this....ASAP​

Answer 1

Answer:

First question: Since this shape is a square, the midpoint of the two diagonals shall coincide with each other.

Second question: assume that by "midpoint" the question refers to the centroid of the triangle. The centroid of a triangle is on its median 2/3 the way from the corresponding vertice.

Step-by-step explanation:

First question

Midpoint of the diagonal between (x1, y1) and (x3, y3):

$\displaystyle \left(\frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2}\right)$.

Similarly, midpoint of the diagonal between (x2, y2) and (x4, y4):

$\displaystyle \left(\frac{x_2 + x_4}{2}, \frac{y_2 + y_4}{2}\right)$.

The two midpoints shall coincide. Therefore,

$\displaystyle \frac{x_1 + x_3}{2} = \frac{x_2 + x_4}{2} \implies x_1 + x_3 = x_2 + x_4$.

Similarly,

$\displaystyle \frac{y_1 + y_3}{2} = \frac{y_2 + y_4}{2} \implies y_1 + y_3 = x_2 + x_4$.

Second question

The centroid of a triangle divides all three of its medians at a 2:1 ratio. If the length of a median of the triangle is $L$, the centroid of that triangle is at a distance of $2/3\;L$ from the vertex on that median.

Start with the median that goes through the vertex $(x_1, y_1)$. That median also goes through the midpoint between $(x_2, y_2)$ and $(x_3, y_3)$.

• Vertex: $(x_1, y_1)$.
• Midpoint of the opposite side: $\displaystyle \left (\frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2}\right)$

The centroid will be located at

$\displaystyle \left (\phantom{\frac{\phantom{x_{1}}}{2}}\right.x_1 + \frac{2}{3}\underbrace{\left(\frac{x_2 + x_3}{2} - x_1\right)}_{\begin{gathered}\text{Separation}\\[-0.5em]\text{in }x\text{-}\\[-0.5em]\text{direction}\end{gathered}}, \quad y_1 + \frac{2}{3}\underbrace{\left(\frac{y_2 + y_3}{2} - y_1\right)}_{{\begin{gathered}\text{Separation}\\[-0.5em]\text{in }y\text{-}\\[-0.5em]\text{direction}\end{gathered}}}\left.\phantom{\frac{\phantom{x_{1}}}{2}}\right)\right)$.

Simplify this expression to obtain:

$\displaystyle \left (\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 +y_2 + y_3}{3}\right)\right)$.