Convert to decimals: \(-\tfrac{1}{2}=-0.5\). On a number line, more negative = smaller.
📘 Explanation
Convert to decimals: \(-0.7,\ -0.5,\ 0,\ 0.75\)
On the number line left→right: \(-0.7 \lt -0.5 \lt 0 \lt 0.75\)
So: \(-0.7 \lt -\tfrac{1}{2} \lt 0 \lt \tfrac{3}{4}\)
Q 03TrapHard
⚠️ Top-tricky: students confuse \((-2)^3\) vs \(-2^3\).
Calculate: \((-2)^3 + (-2^3)\)
📐 Key Rule
\((-2)^3 = (-2)\times(-2)\times(-2) = -8\) \(-2^3 = -(2\times2\times2) = -8\) The parentheses change which value is cubed.
📘 Explanation
\((-2)^3 = -8\) (negative base, odd power → negative)
\(-2^3 = -(8) = -8\) (the minus is outside)
Sum: \(-8 + (-8) = -16\)
Unit 02Linear Equations & Inequalities
⚡ Key MemoryMOVE = FLIP SIGN — moving term across = flips ±. Multiply/divide inequality by negative → FLIP < >
For the line \(y = -3x + 6\), find the x-intercept.
📐 Intercept Rule
x-intercept → set \(y=0\) and solve for \(x\) y-intercept → set \(x=0\) and solve for \(y\)
📘 Explanation
Set \(y=0\): \(0=-3x+6 \Rightarrow 3x=6 \Rightarrow x=2\)
x-intercept is the point where the line crosses the x-axis: \((2,0)\)
Q 10Hard
Which line is parallel to \(y = 2x - 5\)?
📐 Parallel = Same Slope
Two lines are parallel if and only if their slopes are equal. Rearrange to \(y=mx+b\) form to compare slopes.
📘 Explanation
The given line has slope \(m=2\).
Only \(y=2x+7\) also has slope \(m=2\).
(D) rearranges to \(y=\tfrac{1}{2}x-\tfrac{5}{2}\), slope \(\tfrac{1}{2}\) — not parallel.
Unit 04Properties of Figures
⚡ Key MemoryANGLE SUM RULES — Triangle: 180°. Quadrilateral: 360°. Exterior angle of △ = sum of two non-adjacent interior angles.
Q 11Core
In triangle \(ABC\), \(\angle A = 50°\) and \(\angle B = 70°\). Find \(\angle C\).
📘 Explanation
Sum of angles in a triangle = 180°
\(\angle C = 180° - 50° - 70° = 60°\)
Q 12Trap
⚠️ The exterior angle theorem trips up many students!
An exterior angle of a triangle measures 110°. One of the non-adjacent interior angles is 45°. Find the other non-adjacent interior angle.
📐 Exterior Angle Theorem
Exterior angle = sum of the two NON-ADJACENT interior angles \(\text{ext} = \angle A + \angle B\)
📘 Explanation
Exterior angle = 110° = 45° + other angle
Other angle = 110° − 45° = 65°
Q 13Concept
What is the sum of interior angles of a hexagon (6 sides)?
📐 Interior Angle Sum Formula
For an \(n\)-sided polygon: Sum \(= (n-2)\times 180°\)
📘 Explanation
\((6-2) \times 180° = 4 \times 180° = 720°\)
Unit 05Congruence & Similarity
⚡ Key MemorySAS · ASA · SSS · AAS for congruence | AA · SAS · SSS for similarity | Congruent ≅ same size · Similar ~ same shape
Q 14Core
Two triangles are similar with a ratio of \(2:3\). If the shorter triangle has a side of 8 cm, what is the corresponding side in the larger triangle?
📐 Similar Ratio
Set up a proportion: \(\dfrac{2}{3} = \dfrac{8}{x}\) Cross-multiply to solve.
📘 Explanation
\(\dfrac{2}{3} = \dfrac{8}{x}\)
\(2x = 24 \Rightarrow x = 12\) cm
Q 15ConceptTrap
⚠️ Congruence condition — do NOT mix up SSS and SAS!
Which congruence condition is illustrated when two sides and the included angle are known?
📘 Explanation
SAS = Side–Angle–Side: two sides and the angle between them.
The "included" angle means the angle is sandwiched between the two known sides.
Unit 06Mixed Challenge — Most-Missed Problems
⚡ Key MemoryREAD CAREFULLY — Most errors come from rushing. Underline key words. Check your sign. Verify your answer by substituting back.