Created in Canva
A a b C c c a B b Mathematics 10 Summative Test Reviewer • Page 1 MATHEMATICS 10 SUMMATIVE TEST — VISUAL REVIEWER Formulas + diagrams + worked examples + practice C A B SAS / SSS → COSINES | ASA / AAS SINES | BEARINGS CLOCKWISE FROM NORTH QUADRATIC INEQUALITY: Factor Critical Points Test Intervals → Solution ↔ ↔ ↔ MASTER MAP → → → → Exam habit: write the formula first, substitute carefully,thencheck if the answermakessense.
A a b C c c a B b Mathematics 10 Summative Test Reviewer • Page 2 2 — BASICS OF TRIANGLES C A B ↔ ↔ ↔ ANGLE SUM A + B + C = 180° TYPES RIGHT VS. OBLIQUE KEY MATCH A is opposite a, B is opposite b, C is opposite c. Right=one90°angle.Oblique = no 90° angle. Scalene=allsidesdifferent.Isosceles = 2 equal sides. Equilateral = 3 equal sides and 60° each.
A a b C c c a B b Mathematics 10 Summative Test Reviewer • Page 3 3 — LAW OF SINESC A B ↔ ↔ ↔ FORMULA USE IT ASA or AAS. HOW TO SEE IT STEPS ① Match pair MEMORY ② a / sin A = b / sin B = c / sin C ASA/AAS SINES. write proportion cross multiply Matcheachknownangle with the side directly opposite it. solve round/check.③ ④ ⑤→ → → → →
Mathematics 10 Summative Test Reviewer • Page 4 4 — LAW OF SINES: EXAMPLE GIVEN SET UP SOLVE CHECK REAL LIFE Surveying: find an inaccessible distance using angles and a known side. 12/sin42°=b / sin68° b=12(sin68°)/(sin42°) ≈ 16.5 cm A=42°,B=68°, a = 12 cm. Find b. B>A,sobshould be > a. 16.5 > 12 ✓
A a b C c c a B b Mathematics 10 Summative Test Reviewer • Page 5 5 — LAW OF COSINES C A B ↔ ↔ ↔ FORMULA USE IT SAS or SSS. SAS VISUAL STEPS ① Identify a,b,C → MEMORY c² = a² + b² 2ab cos C− SAS COSINES. substitute calculate c² TheangleCmustbe BETWEEN the two known sides a and b. square root.② ③ ④ → → →
Mathematics 10 Summative Test Reviewer • Page 6 6 — LAW OF COSINES: EXAMPLE GIVEN SET UP c² = 7² + 10² − SOLVE c² = 79 → c = √79 ≈ WHY COSINES? REAL LIFE Navigation and route-distance problems. 8.89 m 2(7)(10)cos60° a=7m,b=10m,C=60°. Find c. Twosides+includedangle = SAS.
35° N 35° E = 035° Mathematics 10 Summative Test Reviewer • Page 7 7 — BEARINGS W N S E DEFINITION CARDINALS EXAMPLE N 35° E → FORMAT Always use three digits: 5° 005°, 35° N=000°•E=090° • S = 180° • W = 270° startatNorth, turn 35° toward East 035°. 035°. Abearingisadirection measured clockwise from North. → → →
35° N 35° E = 035° Mathematics 10 Summative Test Reviewer • Page 8 8 — BEARINGS: PRACTICE W N S E BOAT CONVERT N25°E → 025° CONVERT S30°E → 150° CONVERT N70°W → 290° CONVERT S45°W → 225° REMEMBER Start at North and move clockwise. AboattravelsN40° E bearing = 040°.→
Mathematics 10 Summative Test Reviewer • Page 9 9 — QUADRATIC INEQUALITIES WHAT IS IT? SYMBOLS < = less than • > = greater than • ≤ EQUATION VS. INEQUALITY Equation → exact value(s). Inequality → BIG IDEA The sign of the factored expression can change at its critical points. =less/equal • Aninequalitycontainingx²,suchasax²+bx + c < 0. arange of values. = greater/equal≥
critical points Mathematics 10 Summative Test Reviewer • Page 10 10 — FACTORING + CRITICAL POINTS -4 -3 -2 -1 0 1 2 3 4 EXAMPLE x² − 5x + 6 < 0 FACTOR (x − 2)(x − 3) < 0 CRITICAL POINTS WHY? These points split the number line into intervals. Seteachfactortozero:x= 2 and x = 3.
critical points Mathematics 10 Summative Test Reviewer • Page 11 11 — INTERVAL TESTING -4 -3 -2 -1 0 1 2 3 4 INTERVALS TEST Chooseonenumber from each interval and check the sign. RESULT − − ANSWER 2 < x < 3 METHOD Critical points test intervals Criticalpoints2and 3 create: ( ,2), (2,3), (3, ). For(x2)(x3)<0, the middle interval is negative. keep the intervals matching the inequality.→ → −∞ ∞
critical points Mathematics 10 Summative Test Reviewer • Page 12 12 — ENDPOINTS -4 -3 -2 -1 0 1 2 3 4 STRICT < and > → INCLUSIVE ≤ and ≥ → EXAMPLE x² − 9 < 0 → − ANSWER −3 < x < 3 → − WHY? At x = ±3, the expression equals 0, but 0 is not less than 0. ( 3,3) (x3)(x+3) < 0 endpoint included endpoint excluded closed circle / brackets. open circle / parentheses.→ →
Mathematics 10 Summative Test Reviewer • Page 13 13 — SOLUTION NOTATION INEQUALITY −2 < x ≤ 5 INTERVAL (−2,5] SET-BUILDER {x | −2 < x ≤ 5} SYMBOL GUIDE NUMBER-LINE IDEA ( ) = excluded • [ ] = included Opencircle=notincluded.Closed circle = included.
critical points Mathematics 10 Summative Test Reviewer • Page 14 14 — QUADRATIC INEQUALITY EXAMPLES -4 -3 -2 -1 0 1 2 3 4 EXAMPLE 1 x² + 2x − 8 ≥ 0 → (x+4)(x2) EXAMPLE 2 2x² − 7x + 3 > 0 → GARDEN PROBLEM Translate the condition CHECK Lengths and widths are usually positive. (2x1)(x3) > 0 4 or x critical points 2 x < 1/2 or x > 3 test keep physically possible values. − ≥ 0 → x ≤ − − − → →factor → ≥ → →
y shade below Mathematics 10 Summative Test Reviewer • Page 15 x 15 — TWO-VARIABLE QUADRATIC INEQUALITIES EXAMPLE y ≤ −x² + 4 BOUNDARY LINE STYLE or SHADING solid.<or > Graph y = x² + 4 first. shade below. y f(x) dashed. shade above. − ≤ ≥ → y ≤ f(x) → → ≥ →
y shade above Mathematics 10 Summative Test Reviewer • Page 16 x 16 — GRAPHING EXAMPLES EXAMPLE A y ≤ −x² + 4 → EXAMPLE B y ≥ x² → FOUR-RULE MEMORY ≤ = below/solid • ≥ TIP Pick a test point if you are unsure which region to shade. solidparabola+shade above. solidparabola+shade below. =above/solid • < = below/dashed • > = above/dashed
Mathematics 10 Summative Test Reviewer • Page 17 17 — REAL-WORLD APPLICATIONS MOTION AREA Quadraticinequalitiescan restrict possible garden or field dimensions. DOMAIN FINAL CHECK Ask: Does this answer make sense in the situation? Ifxistime,length,oranother physical quantity, reject impossible values. Aquadraticcandescribe when an object is above/below a certain height.
Mathematics 10 Summative Test Reviewer • Page 18 18 — MIXED PRACTICE + ANSWERS 1 — SINES A=30°, B=60°, a=10 → 2 — COSINES a=5, b=8, C=60° → 3 — BEARING N50°E → 050° 4 — QUADRATIC x²−4x+3<0 → 1 < x < 3 5 — QUADRATIC x²−16≥0 → x≤−4 or x≥4 SELF-CHECK Can you explain why each method was chosen? findc. Answer: 7 find b. Answer: 17.32
Mathematics 10 Summative Test Reviewer • Page 19 19 — EXAM TIPS & MNEMONICS TRIANGLES SAS/SSS → BEARINGS QUADRATIC 1 Factor → QUADRATIC 2 <, > exclude. ≤, ≥ GRAPHING Solid for ≤/≥ FAST MEMORY SAS = Cos. ASA/AAS = Sines. ClockwisefromNorth. include. CriticalPoints .Dashedfor </>. COSINES • ASA/AAS Test Intervals SINES Answer→ → →
Mathematics 10 Summative Test Reviewer • Page 20 20 — FINAL ONE-PAGE CHEAT SHEET LAW OF SINES a/sinA = b/sinB = c/sinC → LAW OF COSINES c² = a²+b²−2ab cosC → BEARINGS QUADRATIC 1 Factor → critical points → QUADRATIC 2 <, > = exclude; ≤, ≥ GRAPHING ≤/≥ = solid; </> = dashed; ≤ LAST REMINDER Read the given information first. Then choose the method. = include SAS/SSS =below; ASA/AAS test intervals = above ClockwisefromNorth:N000°,E 090°, S 180°, W 270° ≥