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The midpoint formula helps us find the middle point between two endpoints of a line segment in coordinate geometry. This point is called the midpoint. Imagine a line segment with endpoints at \(B(9, 12)\) and \(D(12, 3)\). To find the midpoint \(M(x, y)\), you should add the x-coordinates together and divide by 2, and do the same with the y-coordinates:

• Calculate \(x\) with \(\frac{9 + 12}{2}\), which results in \(10.5\).
• Calculate \(y\) with \(\frac{12 + 3}{2}\), yielding \(7.5\).

Thus, the midpoint \((x, y)\) is at \( (10.5, 7.5) \). This means point \(F\) with coordinates \((10.5, 7.5)\) is exactly in the middle of the line segment \(\overline{BD}\). This makes it an important reference point when considering other geometrical properties of the triangle.

The slope of a line measures its steepness and is key in understanding the nature of a line in coordinate geometry. For a line between two points \((x_1, y_1)\) and \((x_2, y_2)\), the formula is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

To find the slope of segment \(\overline{BD}\) using points \(B(9, 12)\) and \(D(12, 3)\), follow these steps:

• Subtract the y-coordinates: \(3 - 12 = -9\).
• Subtract the x-coordinates: \(12 - 9 = 3\).
• Calculate the slope: \(\frac{-9}{3} = -3\).

This negative slope tells us that \(\overline{BD}\) slopes downward from left to right. Calculating the slope is essential to find out if two lines are perpendicular, as you'll see later in the perpendicular bisector section.

A line is a perpendicular bisector if it bisects (divides into two equal parts) another line at a right angle (90 degrees). To establish whether \(\overline{EF}\) is a perpendicular bisector of \(\overline{BD}\), two conditions must be met:

• \(\overline{EF}\) must pass through the midpoint of \(\overline{BD}\).
• The lines \(\overline{EF}\) and \(\overline{BD}\) should be perpendicular, which means their slopes' product should be \(-1\).

From prior calculations:

• Point \(F\), matching the midpoint \((10.5, 7.5)\), confirms the first condition.
• The slope of \(\overline{BD}\) is \(-3\) and for \(\overline{EF}\), it is \(\frac{1}{3}\).
• Their slopes' product: \(-3 \times \frac{1}{3} = -1\).

Both conditions are satisfied, confirming \(\overline{EF}\) as a perpendicular bisector of \(\overline{BD}\). This shows its vital role in dividing \(\overline{BD}\) snugly and orthogonally at the midpoint.

Coordinate geometry, or analytic geometry, combines algebra and geometry to study geometrical figures through a coordinate plane. This mathematical method allows precise description and analytical understanding of geometric figures.

• Points are defined through coordinates \((x, y)\).
• Lines are explored using equations and slopes.
• Midpoints and bisectors are identified using formulas.

In the given problem, coordinate geometry helps in identifying key properties of triangle \(\triangle ABC\) and segments \(\overline{BD}\) and \(\overline{EF}\). By leveraging coordinates, one can determine properties like midpoints, slopes, and even slopes’ perpendicular relations smoothly. Exploring shapes and lines using coordinate geometry turns complex geometrical problems into manageable algebraic tasks by providing a visual framework to directly quantify and examine spatial relations.