Miscellaneous Exercise 8: Review of Analytical Geometry

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Chapter Overview & SLOs

What is covered in the Miscellaneous Exercise? The Miscellaneous Exercise for Chapter 8 provides a comprehensive review of analytical geometry, including line equations, slopes, angles, intersections, and geometric proofs.

How do we convert between different forms of a line? Transform general equations into other standard forms:

• General form: $Ax + By + C = 0$
• Slope-intercept form: $y = mx + c$ → isolate $y$
• Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$ → express $x$ and $y$ intercepts

Example - Converting general to slope-intercept: $5x - 2y + 1 = 0$

• $5x - 2y + 1 = 0$ → $-2y = -5x - 1$ → $y = \frac{5}{2}x + \frac{1}{2}$

Example - Converting to intercept form: From $5x - 2y + 1 = 0$:

• $5x - 2y = -1$ → Divide by $-1$: $-5x + 2y = 1$
• Or directly: $\frac{x}{-1/5} + \frac{y}{1/2} = 1$

How do we convert normal form to general form? Normal form: $x \cos \theta + y \sin \theta = p$

• Rearrange to $x \cos \theta + y \sin \theta - p = 0$
• Here $A = \cos \theta$, $B = \sin \theta$, $C = -p$

How do we prove collinearity using slopes? Three points $A$, $B$, and $C$ are collinear if:

• $m_{AB} = m_{AC}$ (or $m_{AB} = m_{BC}$)
• Example: $A(1,2)$, $B(3,4)$, $C(5,6)$ → $m_{AB} = (4-2)/(3-1) = 2/2 = 1$, $m_{AC} = (6-2)/(5-1) = 4/4 = 1$ → collinear

How do we find interior angles of a triangle? For triangle $ABC$, find slopes of sides:

• $\angle A$ uses slopes $m_{AB}$ and $m_{AC}$
• $\angle B$ uses slopes $m_{BA}$ and $m_{BC}$
• $\angle C$ uses slopes $m_{CA}$ and $m_{CB}$
• Formula: $\tan \alpha = \left|\frac{m_j - m_i}{1 + m_j m_i}\right|$ gives acute angle
• If the calculated $\tan \theta$ is negative, the angle is obtuse: $\theta = 180^\circ - |\text{acute angle}|$

Example - Interior angles: For triangle $A(1,2)$, $B(4,6)$, $C(7,2)$:

• $m_{AB} = 4/3$, $m_{AC} = 0$, $m_{BC} = -4/3$
• $\tan \angle A = \left|(0 - 4/3)/(1 + 0)\right| = 4/3$ → $\angle A = 53.13^\circ$
• $\tan \angle B = \left|(-4/3 - 4/3)/(1 + (-4/3)(4/3))\right| = \left|(-8/3)/(1 - 16/9)\right| = \left|(-8/3)/(-7/9)\right| = 24/7$ → $\angle B = 73.74^\circ$
• $\angle C = 180^\circ - 53.13^\circ - 73.74^\circ = 53.13^\circ$

How do we find the point of intersection of two lines? Solve simultaneous equations using elimination or substitution.

• Example: Find intersection of $OC$ and $AB$ given their equations
• Result: $(\frac{155}{63}, \frac{62}{21})$

How do we derive equations of parallel and perpendicular lines? Use point-slope form $y - y_1 = m(x - x_1)$:

• Parallel: $m_2 = m_1$
• Perpendicular: $m_2 = -1/m_1$
• Example: Line through $(2,3)$ parallel to line with slope $4$ → $y - 3 = 4(x - 2)$ → $y = 4x - 5$

How do we find line through midpoint? First find midpoint using $M = ((x_1+x_2)/2, (y_1+y_2)/2)$, then use point-slope form with given slope condition.

Quadrant adjustment for angles: If $\tan \theta$ is negative, the angle is in Quadrant II (between $90^\circ$ and $180^\circ$). The interior angle of a triangle can be obtuse, so $\theta_{\text{interior}} = 180^\circ - \tan^{-1}(|\text{value}|)$.

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

• How do we transform between linear equation forms? Master the conversion of lines from general form ($Ax + By + C = 0$) to slope-intercept form ($y = mx + c$), intercept form ($\frac{x}{a} + \frac{y}{b} = 1$), and normal form ($x\cos\theta + y\sin\theta = p$), and vice versa.
• How do we apply slope properties to geometric proofs? Use equal slopes ($m_1 = m_2$) to prove collinearity of three points, and use negative reciprocal slopes ($m_1 \times m_2 = -1$) to prove perpendicular lines and find equations of altitudes.
• How do we calculate angles within geometric figures? Apply the tangent formula $\tan \alpha = \frac{m_j - m_i}{1 + m_j m_i}$ to the slopes of intersecting lines to find specific interior angles of triangles, adjusting for obtuse angles by using $180^\circ - |\text{acute}|$ when the result is negative.
• How do we solve for intersections and midpoints? Solve simultaneous linear equations using elimination or substitution to locate intersection points of lines, and use the midpoint formula $M = ((x_1+x_2)/2, (y_1+y_2)/2)$ to find centers of segments for navigation and geometric modeling.

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