Miscellaneous Exercise 7: Review of Coordinate Geometry

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Chapter Overview & SLOs

What is covered in the Miscellaneous Exercise? The Miscellaneous Exercise for Chapter 7 provides a comprehensive review of coordinate geometry, including distance formula, midpoint formula, geometric proofs, and quadrant analysis.

How do we prove a quadrilateral is a parallelogram? A quadrilateral is a parallelogram if both pairs of opposite sides are equal AND parallel.

• Equal sides: Use distance formula to show $AB = CD$ and $BC = DA$
• Parallel sides: Use slope formula to show slopes of $AB$ and $CD$ are equal, and slopes of $BC$ and $DA$ are equal
• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

How do we find missing coordinates from midpoint data? When a point is the midpoint of two points, set up equations using the midpoint formula.

• Example: $B(-1,7)$ is the midpoint of $A(3,4)$ and $C(x,y)$
• $(-1,7) = \left(\frac{3 + x}{2}, \frac{4 + y}{2}\right)$
• $-1 = \frac{3 + x}{2}$ → $3 + x = -2$ → $x = -5$
• $7 = \frac{4 + y}{2}$ → $4 + y = 14$ → $y = 10$
• Therefore, $C(-5, 10)$

How do we find the center of a circle from diameter endpoints? The center of a circle is the midpoint of any diameter.

• Example: Diameter endpoints are $A(4,5)$ and $B(6,9)$
• Center $M = \left(\frac{4+6}{2}, \frac{5+9}{2}\right) = \left(\frac{10}{2}, \frac{14}{2}\right) = (5,7)$

How do we determine the quadrant of a point? Quadrants are determined by the signs of $x$ and $y$ coordinates:

• Quadrant I: $x > 0$, $y > 0$
• Quadrant II: $x < 0$, $y > 0$
• Quadrant III: $x < 0$, $y < 0$
• Quadrant IV: $x > 0$, $y < 0$
• Example: $(-4, -(-6)) = (-4, 6)$ → $x < 0$, $y > 0$ → Quadrant II

How do we calculate the hypotenuse of a triangle? The hypotenuse is the distance between the endpoints of the longest side (or the side opposite the right angle).

• Example: For vertices $A(2,3)$, $B(5,7)$, $C(2,10)$
• $AB = \sqrt{(5-2)^2 + (7-3)^2} = \sqrt{9 + 16} = 5$
• $BC = \sqrt{(2-5)^2 + (10-7)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$
• $AC = \sqrt{(2-2)^2 + (10-3)^2} = \sqrt{0 + 49} = 7$
• If $5^2 + (3\sqrt{2})^2 = 25 + 18 = 43 \neq 49$, not a right triangle. The largest side is $7$.

What is the abscissa on the y-axis? The abscissa (x-coordinate) of any point on the y-axis is always $0$.

How do we find the intersection of diagonals in a kite or square? The intersection point of the diagonals is the midpoint of both diagonals.

• In a square, diagonals bisect each other at the center
• Use midpoint formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

How do we find missing coordinates when distance is given? Use the distance formula with one coordinate unknown, then solve the resulting equation.

• Example: Distance between $P(2,3)$ and $Q(x,7)$ is $5$ units
• $\sqrt{(x-2)^2 + (7-3)^2} = 5$
• $(x-2)^2 + 16 = 25$ → $(x-2)^2 = 9$ → $x-2 = \pm 3$ → $x = 5$ or $x = -1$

Key formulas to remember:

• Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Midpoint formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

These solutions are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we apply the distance formula to find lengths? Use $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ to compare segment lengths, calculate triangle hypotenuses, find values of missing coordinates when distance is given (e.g., $20$ units), and prove geometric properties like equal sides in parallelograms.
• How do we utilize the midpoint formula for geometric centers? Calculate the center of a circle from diameter endpoints, the intersection of diagonals in quadrilaterals (kite, square, rectangle), and find missing vertices when a midpoint is given, by averaging the $x$ and $y$ coordinates of endpoints using $M = ((x_1+x_2)/2, (y_1+y_2)/2)$.
• How do we identify points in the coordinate plane? Determine the quadrant based on signs (e.g., $x<0, y>0$ for Quadrant II), understand that the abscissa on the y-axis is always zero, and recognize that the minimum distance from a point to an axis is determined by the absolute value of its coordinates.
• How do we verify geometric shapes? Prove properties of parallelograms by checking for equal and parallel opposite sides using distance and slope formulas, and identify triangle types (right, isosceles, equilateral, scalene) by comparing the squares of their side lengths and checking Pythagorean theorem.

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