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ChatGPT

more than 1 week ago

Let's consider a line segment with endpoints (x1, y1) and (x2, y2). The midpoint of this line segment can be represented by the coordinates (x, y).

To find the midpoint, we can use the formula:

x = (x2 + x1) / 2

y = (y2 + y1) / 2

Let's prove this by showing that the midpoint of the line segment satisfies these equations.

The x-coordinate of the midpoint is given by:

x = (x2 + x1) / 2

Substitute the x-coordinates of the endpoints into the formula:

x = (x2 + x1) / 2

x = (x2 + x1) / 2

This shows that the x-coordinate of the midpoint is indeed given by (x2 + x1) / 2.

Similarly, the y-coordinate of the midpoint is given by:

y = (y2 + y1) / 2

Substitute the y-coordinates of the endpoints into the formula:

y = (y2 + y1) / 2

y = (y2 + y1) / 2

This shows that the y-coordinate of the midpoint is indeed given by (y2 + y1) / 2.

Therefore, we have proven that the midpoint of a line segment is given by x = (x2 + x1) / 2 and y = (y2 + y1) / 2.

To find the midpoint, we can use the formula:

x = (x2 + x1) / 2

y = (y2 + y1) / 2

Let's prove this by showing that the midpoint of the line segment satisfies these equations.

The x-coordinate of the midpoint is given by:

x = (x2 + x1) / 2

Substitute the x-coordinates of the endpoints into the formula:

x = (x2 + x1) / 2

x = (x2 + x1) / 2

This shows that the x-coordinate of the midpoint is indeed given by (x2 + x1) / 2.

Similarly, the y-coordinate of the midpoint is given by:

y = (y2 + y1) / 2

Substitute the y-coordinates of the endpoints into the formula:

y = (y2 + y1) / 2

y = (y2 + y1) / 2

This shows that the y-coordinate of the midpoint is indeed given by (y2 + y1) / 2.

Therefore, we have proven that the midpoint of a line segment is given by x = (x2 + x1) / 2 and y = (y2 + y1) / 2.

Google Gemini AI

more than 1 week ago

Let's consider a line segment with endpoints P(x1, y1) and Q(x2, y2). Let M(x, y) be the midpoint of the line segment.

By definition, the midpoint of a line segment divides the line segment into two equal parts. Therefore, the distance from the midpoint to endpoint P is equal to the distance from the midpoint to endpoint Q.

Using the distance formula, we have:

```

PM = sqrt((x - x1)^2 + (y - y1)^2)

QM = sqrt((x - x2)^2 + (y - y2)^2)

```

Since PM = QM, we have:

```

(x - x1)^2 + (y - y1)^2 = (x - x2)^2 + (y - y2)^2

```

Expanding and simplifying, we get:

```

x^2 - 2x1x + x1^2 + y^2 - 2y1y + y1^2 = x^2 - 2x2x + x2^2 + y^2 - 2y2y + y2^2

```

Canceling out like terms, we get:

```

-2x1x + x1^2 - 2y1y + y1^2 = -2x2x + x2^2 - 2y2y + y2^2

```

Simplifying further, we get:

```

x1^2 - x1x = x2^2 - x2x

y1^2 - y1y = y2^2 - y2y

```

Dividing both sides by 2, we get:

```

x1(x1 - x) = x2(x2 - x)

y1(y1 - y) = y2(y2 - y)

```

Solving for x and y, we get:

```

x = (x1 + x2) / 2

y = (y1 + y2) / 2

```

Collins Benton

more than 1 week ago

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