Without using distance formula, show that points $(-2,-1)$, $(4,0)$, $(3,3)$ and $(-3,2)$ are the vertices of a Parallelogram.

**OR**

The line through the points $(h, 3)$ and $(4, 1)$ intersects the line $7x - 9y - 19 = 0$ at right angles. Find the value of $h$.

Question

Without using distance formula, show that points $(-2,-1)$, $(4,0)$, $(3,3)$ and $(-3,2)$ are the vertices of a Parallelogram.

**OR**

The line through the points $(h, 3)$ and $(4, 1)$ intersects the line $7x - 9y - 19 = 0$ at right angles. Find the value of $h$.

Accepted Answer

>

> [!TLDR]
> The diagonals $AC$ and $BD$ share the same midpoint, so the diagonals bisect each other — a property unique to parallelograms.
>
> Let $A(-2,-1)$, $B(4,0)$, $C(3,3)$, $D(-3,2)$ be the points taken in order.
>
> **Key fact (no distance formula needed):** a quadrilateral is a parallelogram if and only if its diagonals bisect each other, i.e. the midpoint of one diagonal equals the midpoint of the other.
>
> Midpoint formula: for points $(x_1,y_1)$ and $(x_2,y_2)$, the midpoint is $\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$.
>
> **Midpoint of diagonal $AC$:**
> $$\left(\dfrac{-2+3}{2},\dfrac{-1+3}{2}\right)=\left(\dfrac12,\ 1\right).$$
>
> **Midpoint of diagonal $BD$:**
> $$\left(\dfrac{4+(-3)}{2},\dfrac{0+2}{2}\right)=\left(\dfrac12,\ 1\right).$$
>
> Both diagonals have the same midpoint $\left(\dfrac12,1\right)$, so they bisect each other. Therefore $ABCD$ is a parallelogram.
> …

Slope value	What the line does
$m > 0$	Rises left to right (uphill)
$m < 0$	Falls left to right (downhill)
$m = 0$	Horizontal (flat)
$m$ undefined	Vertical (straight up/down)

Q.Without using distance formula, show that points $(-2,-1)$ , $(4,0)$ , $(3,3)$ and $(-3,2)$ are the vertices of a Parallelogram. OR The line through the points $(h, 3)$ and $(4, 1)$ intersects the line $7x - 9y - 19 = 0$ at right angles. Find the value of $h$ .

Concept understanding — Slope Calculation

Slope Calculation — From Intuition to Precision

The Intuition: Rise Over Run

The Precise Definition

What the Number Tells You

A Worked Example

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