LFM Pure

brazil-enem 2010 Q141 Find an angle using the cosine rule View

Question 141
A figura representa um mapa com a localização de três cidades: A, B e C. As distâncias entre as cidades, em linha reta, são: A a B = 80 km, B a C = 60 km e A a C = 100 km.
[Figure]
O ângulo formado pelas estradas AB e AC, em graus, é
(A) $30^\circ$ (B) $37^\circ$ (C) $45^\circ$ (D) $53^\circ$ (E) $60^\circ$

brazil-enem 2010 Q169 Area Computation in Coordinate Geometry View

Question 169
A figura mostra dois triângulos semelhantes $ABC$ e $DEF$.
[Figure]
Se $AB = 6$ cm, $BC = 8$ cm, $AC = 10$ cm e $DE = 9$ cm, o perímetro do triângulo $DEF$, em cm, é
(A) 24 (B) 30 (C) 36 (D) 40 (E) 45

brazil-enem 2011 Q164 Compute area of a triangle or related figure View

Um triângulo tem lados medindo 5 cm, 12 cm e 13 cm. A área desse triângulo é
(A) 20 cm$^2$ (B) 25 cm$^2$ (C) 30 cm$^2$ (D) 35 cm$^2$ (E) 40 cm$^2$

brazil-enem 2024 Q152 Compute area of a triangle or related figure View

A triangle has sides measuring 5 cm, 12 cm, and 13 cm. What is the area, in square centimeters, of this triangle?
(A) 20
(B) 25
(C) 30
(D) 35
(E) 40

cmi-entrance 2014 QA10 4 marks Ambiguous case and triangle existence/uniqueness View

In each of the following independent situations we want to construct a triangle $ABC$ satisfying the given conditions. In each case state how many such triangles $ABC$ exist up to congruence.
(i) $AB = 30 \quad BC = 95 \quad AC = 55$
(ii) $\angle A = 30 ^ { \circ } \quad \angle B = 95 ^ { \circ } \quad \angle C = 55 ^ { \circ }$
(iii) $\angle A = 30 ^ { \circ } \quad \angle B = 95 ^ { \circ } \quad BC = 55$
(iv) $\angle A = 30 ^ { \circ } \quad AB = 95 \quad BC = 55$

cmi-entrance 2018 QB4 15 marks Multi-step composite figure problem View

Let $ABC$ be an equilateral triangle with side length 2. Point $A'$ is chosen on side $BC$ such that the length of $A'B$ is $k < 1$. Likewise points $B'$ and $C'$ are chosen on sides $CA$ and $AB$ with $CB' = AC' = k$. Line segments are drawn from points $A', B', C'$ to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled $PQR$. Show that $PQR$ is an equilateral triangle with side length $\dfrac{4(1-k)}{\sqrt{k^{2}-2k+4}}$.

cmi-entrance 2021 Q3 4 marks Ambiguous case and triangle existence/uniqueness View

We want to construct a triangle ABC such that angle A is $20.21 ^ { \circ }$, side AB has length 1 and side BC has length $x$ where $x$ is a positive real number. Let $N ( x ) =$ the number of pairwise noncongruent triangles with the required properties.
(a) There exists a value of $x$ such that $N ( x ) = 0$.
(b) There exists a value of $x$ such that $N ( x ) = 1$.
(c) There exists a value of $x$ such that $N ( x ) = 2$.
(d) There exists a value of $x$ such that $N ( x ) = 3$.

csat-suneung 2008 Q28 3 marks Circumradius or incircle radius computation View

(Calculus) As shown in the figure, for a positive angle $\theta$, there is an isosceles triangle ABC with $\angle \mathrm { ABC } = \angle \mathrm { ACB } = \theta$ and $\overline { \mathrm { BC } } = 2$. Let O be the center of the inscribed circle of triangle ABC, D be the point where segment AB meets the inscribed circle, and E be the point where segment AC meets the inscribed circle. [3 points]
(1) $\frac { \pi } { 4 } - 1$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 4 } + \frac { 1 } { 3 }$
(4) $\frac { \pi } { 4 } + \frac { 1 } { 2 }$
(5) $\frac { \pi } { 4 } + 1$

csat-suneung 2013 Q29 4 marks Multi-step composite figure problem View

In triangle ABC, $\overline { \mathrm { AB } } = 1$, $\angle \mathrm { A } = \theta$, and $\angle \mathrm { B } = 2 \theta$. Point D on side AB is chosen so that $\angle \mathrm { ACD } = 2 \angle \mathrm { BCD }$. When $\lim _ { \theta \rightarrow + 0 } \frac { \overline { \mathrm { CD } } } { \theta } = a$, find the value of $27 a ^ { 2 }$. (Given that $0 < \theta < \frac { \pi } { 4 }$.) [4 points]

csat-suneung 2014 Q28 4 marks Limit evaluation involving derivatives or asymptotic analysis View

As shown in the figure, there is an isosceles triangle ABC with AB as one side of length 4, $\overline { \mathrm { AC } } = \overline { \mathrm { BC } }$, and $\angle \mathrm { ACB } = \theta$. On the extension of segment AB, a point D is taken such that $\overline { \mathrm { AC } } = \overline { \mathrm { AD } }$, and a point P is taken such that $\overline { \mathrm { AC } } = \overline { \mathrm { AP } }$ and $\angle \mathrm { PAB } = 2 \theta$. Let $S ( \theta )$ be the area of triangle BDP. Find the value of $\lim _ { \theta \rightarrow + 0 } ( \theta \times S ( \theta ) )$. (Here, $0 < \theta < \frac { \pi } { 6 }$) [4 points]

csat-suneung 2018 Q14 4 marks Addition/Subtraction Formula Evaluation View

As shown in the figure, in triangle ABC with $\overline { \mathrm { AB } } = 5 , \overline { \mathrm { AC } } = 2 \sqrt { 5 }$, let D be the foot of the perpendicular from vertex A to segment BC.
For point E that divides segment AD internally in the ratio $3 : 1$, we have $\overline { \mathrm { EC } } = \sqrt { 5 }$. If $\angle \mathrm { ABD } = \alpha , \angle \mathrm { DCE } = \beta$, what is the value of $\cos ( \alpha - \beta )$? [4 points]
(1) $\frac { \sqrt { 5 } } { 5 }$
(2) $\frac { \sqrt { 5 } } { 4 }$
(3) $\frac { 3 \sqrt { 5 } } { 10 }$
(4) $\frac { 7 \sqrt { 5 } } { 20 }$
(5) $\frac { 2 \sqrt { 5 } } { 5 }$

csat-suneung 2021 Q10 3 marks Circumradius or incircle radius computation View

In triangle ABC with $\angle \mathrm { A } = \frac { \pi } { 3 }$ and $\overline { \mathrm { AB } } : \overline { \mathrm { AC } } = 3 : 1$, the radius of the circumcircle of triangle ABC is 7. What is the length of segment AC? [3 points]
(1) $2 \sqrt { 5 }$
(2) $\sqrt { 21 }$
(3) $\sqrt { 22 }$
(4) $\sqrt { 23 }$
(5) $2 \sqrt { 6 }$

csat-suneung 2021 Q28 4 marks Circumradius or incircle radius computation View

In triangle ABC, $\angle \mathrm { A } = \frac { \pi } { 3 }$ and $\overline { \mathrm { AB } } : \overline { \mathrm { AC } } = 3 : 1$. If the circumradius of triangle ABC is 7, let $k$ be the length of segment AC. Find the value of $k ^ { 2 }$. [4 points]

csat-suneung 2023 Q11 4 marks Circumradius or incircle radius computation View

As shown in the figure, quadrilateral ABCD is inscribed in a circle and $$\overline { \mathrm { AB } } = 5 , \overline { \mathrm { AC } } = 3 \sqrt { 5 } , \overline { \mathrm { AD } } = 7 , \angle \mathrm { BAC } = \angle \mathrm { CAD }$$ What is the radius of this circle? [4 points]
(1) $\frac { 5 \sqrt { 2 } } { 2 }$
(2) $\frac { 8 \sqrt { 5 } } { 5 }$
(3) $\frac { 5 \sqrt { 5 } } { 3 }$
(4) $\frac { 8 \sqrt { 2 } } { 3 }$
(5) $\frac { 9 \sqrt { 3 } } { 4 }$

csat-suneung 2024 Q13 4 marks Multi-step composite figure problem View

As shown in the figure, $$\overline{\mathrm{AB}} = 3, \quad \overline{\mathrm{BC}} = \sqrt{13}, \quad \overline{\mathrm{AD}} \times \overline{\mathrm{CD}} = 9, \quad \angle\mathrm{BAC} = \frac{\pi}{3}$$ For quadrilateral ABCD, let $S_1$ denote the area of triangle ABC, $S_2$ denote the area of triangle ACD, and $R$ denote the circumradius of triangle ACD. If $S_2 = \frac{5}{6}S_1$, find the value of $\frac{R}{\sin(\angle\mathrm{ADC})}$. [4 points]
(1) $\frac{54}{25}$
(2) $\frac{117}{50}$
(3) $\frac{63}{25}$
(4) $\frac{27}{10}$
(5) $\frac{72}{25}$

csat-suneung 2025 Q14 4 marks Multi-step composite figure problem View

As shown in the figure, in triangle ABC, point D is taken on segment AB such that $\overline{\mathrm{AD}} : \overline{\mathrm{DB}} = 3 : 2$, and a circle $O$ centered at A passing through D intersects segment AC at point E. $\sin A : \sin C = 8 : 5$, and the ratio of the areas of triangles ADE and ABC is $9 : 35$. When the circumradius of triangle ABC is 7, what is the maximum area of triangle PBC for a point P on circle $O$? (Given: $\overline{\mathrm{AB}} < \overline{\mathrm{AC}}$) [4 points]
(1) $18 + 15\sqrt{3}$
(2) $24 + 20\sqrt{3}$
(3) $30 + 25\sqrt{3}$
(4) $36 + 30\sqrt{3}$
(5) $42 + 35\sqrt{3}$

csat-suneung 2026 Q18 3 marks Compute area of a triangle or related figure View

In triangle ABC, $\overline { \mathrm { AB } } = 5$, $\overline { \mathrm { AC } } = 6$, and $\cos ( \angle \mathrm { BAC } ) = - \frac { 3 } { 5 }$. Find the area of triangle ABC. [3 points]

gaokao 2010 Q18 Find an angle using the cosine rule View

18. If the three interior angles of $\triangle A B C$ satisfy $\sin A : \sin B : \sin C = 5 : 11 : 13$ , then $\triangle A B C$
A. must be an acute triangle
B. must be a right triangle
C. must be an obtuse triangle
D. could be either an acute triangle or an obtuse triangle

III. Solution Problems (Total Score: 74 points)

gaokao 2011 Q8 Find a side or angle using the sine rule View

8. At two points $A$ and $B$ that are 2 kilometers apart, target $C$ is measured. If $\angle CAB = 75°, \angle CBA = 60°$, then the distance between points $A$ and $C$ is $\_\_\_\_$ kilometers.

gaokao 2015 Q11 5 marks Find a side or angle using the sine rule View

In $\triangle ABC$, $a = 3$, $b = \sqrt { 6 }$, $\angle A = \frac { 2 \pi } { 3 }$, then $\angle B =$

gaokao 2015 Q12 Determine an angle or side from a trigonometric identity/equation View

12. In $\triangle A B C$, $a = 4 , b = 5 , c = 6$, then $\frac { \sin 2 A } { \sin C } =$ $\_\_\_\_$.

gaokao 2015 Q12 Find a side length using the cosine rule View

12. If the area of acute triangle $A B C$ is $10 \sqrt { 3 }$, and $A B = 5$, $A C = 8$, then $B C$ equals $\_\_\_\_$.

gaokao 2015 Q13 5 marks Find a side length using the cosine rule View

In $\triangle A B C$, let the sides opposite to angles $\mathrm { A } , \mathrm { B } , \mathrm { C }$ be $a , b , c$ respectively. Given $a = 2 , \cos C = - \frac { 1 } { 4 } , 3 \sin A = 2 \sin B$, then $\mathrm { c } =$ $\_\_\_\_$ .

gaokao 2015 Q13 Multi-step composite figure problem View

13. In $\triangle \mathrm { ABC }$, $\mathrm { B } = 120 ^ { \circ } , \mathrm { AB } = \sqrt { 2 }$, and the angle bisector from $A$ is $\mathrm { AD } = \sqrt { 3 }$, then $\mathrm { AC } = $ $\_\_\_\_$ . Note for Candidates: Questions (14), (15), and (16) are optional. Please choose any two to answer. If all three are answered, only the first two will be graded.

gaokao 2015 Q13 Heights and distances / angle of elevation problem View

13. As shown in the figure, a car is traveling due west on a horizontal road. At point $A$, the mountain peak $D$ on the north side of the road is measured to be in the direction $30°$ west of north. After traveling 600 m to reach point $B$, the peak is measured to be in the direction $75°$ west of north with an elevation angle of $30°$. The height of the mountain $CD = $ $\_\_\_\_$ m.
[Figure]
Figure for Question 13
[Figure]
Figure for Question 14

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

III. Solution Problems (Total Score: 74 points)

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164