Geometry and Trigonometry — Free Maths AA HL Practice Questions

The perpendicular bisector of $AB$ is the line $L_2$. Find the equation of $L_2$ in the form $y = mx + c$. [3 marks]

The line $L$ intersects the $y$-axis at point $B$ and the $x$-axis at point $A$. Calculate the area of triangle $OAB$, where $O$ is the origin. [3 marks]

Find the perpendicular distance from $O$ to $L$ and verify that is consistent with the area found in part (b). [3 marks]

Determine the equation of $L_2$ in the form $y = mx + c$. The lines $L_1$ and $L_2$ intersect at point $C$. [2 marks]

Determine the coordinates of $C$, giving your answer in exact form. [2 marks]

Determine the ratio $AC : BC$, giving your answer in its simplest integer form. [2 marks]

Find the coordinates of the $x$-intercept of $L$. [1 mark]

A line $M$ is perpendicular to $L$ and passes through the point $P(0,\,5)$. (i) Find the equation of $M$ in the form $y = mx + c$. [2]
(ii) Find the coordinates of the point of intersection of $L$ and $M$. [2 marks]

Using your answer to part (a) and an appropriate compound-angle identity, find the exact value of $\cos\left(\dfrac{5\pi}{3}\right)$. [3 marks]

Hence find the first time $t > 0$ at which the particle is at $O$, giving your answer in the form $a + \dfrac{b}{\pi}\arctan(k)$, where $a, b, k$ are constants to be determined. [2 marks]

The surveyor now considers a general scenario in which the angle of elevation from $A$ is $\alpha$ radians, where $0 < \alpha < \dfrac{\pi}{3}$, the angle from $B$ remains $\dfrac{\pi}{3}$, and the distance $AB$ remains $120$ m. Show that the height of the cliff is given by $$h = \frac{120\sqrt{3}\tan\alpha}{\sqrt{3} - \tan\alpha}$$ and hence deduce, with justification, whether $h$ is an increasing or decreasing function of $\alpha$ on the given domain. [4 marks]

Find the value of $\theta$ in radians. [2 marks]

Find the exact value of $\sin\!\left(\theta + \dfrac{3\pi}{4}\right)$. [2 marks]

Write down the horizontal distance from $Q$ to the base of the cliff in terms of $h$. [1 mark]

Hence, find the value of $h$, giving your answer in the form $a + b\sqrt{3}$ where $a, b \in \mathbb{Z}$. [2 marks]

Using the identity $\sin 68^\circ = \cos 22^\circ$ and the result from part (a), show that $PR = \dfrac{10\sin 42^\circ}{\cos 22^\circ}$. [1 mark]

Hence, using an appropriate double-angle identity, show that $PR = \dfrac{5\sin 42^\circ}{\sin 11^\circ \cos 11^\circ}$, and deduce an expression for $PR$ in the form $\dfrac{k\sin 42^\circ}{\sin 22^\circ}$, stating the value of $k$. [2 marks]

For each value of $\angle ACB$ found in part (a), determine the corresponding length of $AC$, giving your answers in cm correct to three significant figures. [2 marks]

Analyse whether the two possible triangles have the same area. Support your answer with appropriate calculations. [2 marks]

Find the two possible values of angle $B$, correct to $1$ decimal place. [2 marks]

For the case in which triangle $ABC$ is acute, find the exact length of side $c$. [3 marks]

A second line $L_2$ is perpendicular to $L_1$ and passes through the midpoint $M$ of $AB$. Determine the equation of $L_2$ in the form $y = mx + c$. [2 marks]

Find the coordinates of the point of intersection of $L_1$ and $L_2$, and hence verify that this point is $M$. [2 marks]

Determine the equation of line $PQ$ in the form $ax + by = c$, where $a,b,c \in \mathbb{Z}$. [2 marks]

The aircraft's position at time $t$ seconds is given by $$\mathbf{r}(t) = \begin{pmatrix} 100 \\ 200 \end{pmatrix} + t\begin{pmatrix} 15 \\ 30 \end{pmatrix}.$$ Two radio beacons are located at $A(700,\,1400)\,\text{m}$ and $B(1150,\,1700)\,\text{m}$. Determine the time $t$ at which the aircraft is equidistant from $A$ and $B$. [2 marks]

A maintenance hut is located at point $C(60, k)$ on the line $AB$. Determine the value of $k$. [2 marks]

A second resort claims its ski lift, running along the line $y = -\dfrac{2}{3}x + \dfrac{190}{3}$, passes through the base station $A(20, 50)$ and is perpendicular to lift $AB$. A skier starts at $A$ and travels along this second lift until reaching the point $D$ directly above $C$ (i.e. $D$ has the same $x$-coordinate as $C$). Determine the length $AD$ and hence evaluate whether $D$ lies closer to $A$ than $C$ does. [2 marks]

Find the coordinates of the midpoint $M$ of side $AC$. [1 mark]

Calculate the area of triangle $BCM$ with vertices $B(8,1)$, $C(6,9)$, $M(4,7)$, giving your answer in $\text{km}^2$. [2 marks]

Hence, deduce the area of triangle $ABM$ and explain what this result reveals about the median $BM$ of triangle $ABC$. Formulae provided: $$d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$ $$\text{Midpoint} = \left(\frac{x_1+x_2}{2},\,\frac{y_1+y_2}{2}\right)$$ $$\text{Area} = \tfrac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$$ [1 mark]

Calculate the length of the cable to anchor point $B$. Give your answer correct to 3 significant figures. [2 marks]

Given that $A$ and $B$ lie on opposite sides of the mast, calculate the distance $AB$. Give your answer correct to 3 significant figures. [2 marks]

Hence, determine the value of $x$. Give your answer correct to 3 significant figures. [2 marks]

Calculate the straight-line distance from point $A$ to the top of the tower $T$. Give your answer correct to 3 significant figures. [2 marks]

Calculate the first time at which the rider is $34$ metres above the ground. Give your answer correct to 3 significant figures. [4 marks]

Hence show that $x = \dfrac{50\tan 35°}{\tan 52° - \tan 35°}$. [2 marks]

Calculate the height of the tower, giving your answer in metres correct to 3 significant figures. [2 marks]

Calculate the height $h$ of the cliff, giving your answer correct to 3 significant figures. [4 marks]

Calculate the distance $AB$, giving your answer correct to 3 significant figures. [2 marks]

A lighthouse $L$ is located such that is equidistant from $O$, $A$, and $B$. Determine the bearing of $L$ from $O$, correct to the nearest degree. [2 marks]

Hence, use the law of sines to find $\angle ACB$, giving your answer correct to 1 decimal place. [2 marks]

Determine the length of the path $BM$, giving your answer correct to 3 significant figures. [2 marks]

Hence, determine the distance $AB$. Give your answer correct to 3 significant figures. [3 marks]

(i) Find the equation of $L_2$ in the form $ax + by + c = 0$, where $a, b, c \in \mathbb{Z}$. [3 marks]
(ii) Find the coordinates of the point $D$ where $L_1$ and $L_2$ intersect. [2 marks]

$L_3$ intersects $L_1$ at point $F$. Determine the coordinates of $F$ and hence show that $D$, $C$, and $F$ are not collinear. [2 marks]

A triangle has vertices $P(p_1,p_2)$, $Q(q_1,q_2)$, and $R(r_1,r_2)$. The midpoints of $PQ$, $QR$, and $RP$ are $L$, $M$, and $N$ respectively. Prove that the area of triangle $LMN$ is one quarter of the area of triangle $PQR$. The following formulae may be used: - *Midpoint of $(x_1,y_1)$ and $(x_2,y_2)$: $\left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2}\right)$* - *Distance: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$* - *Area of triangle with vertices $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$: $\dfrac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$* [4 marks]

Using the result from (a), evaluate $$\lim_{x \to 0} \frac{1 - \cos x}{x^2},$$ justifying each step of your reasoning. $\quad [6 marks]

A particle $P$ moves on the unit circle with position at time $t$ seconds given by $$P(t)=\left(\cos\!\left(\frac{\pi}{3}t\right), \sin\!\left(\frac{\pi}{3}t\right)\right).$$ A second particle $Q$ moves on the same circle with position $$Q(t)=\left(\cos\!\left(\frac{\pi}{3}t+\frac{\pi}{6}\right), \sin\!\left(\frac{\pi}{3}t+\frac{\pi}{6}\right)\right).$$ (i) Determine the value of $t$ in the interval $0 \leq t \leq 3$ for which the $x$-coordinates of $P$ and $Q$ are equal. Justify your reasoning. [3 marks]
(ii) Show that the distance between $P(t)$ and $Q(t)$ is constant for all $t$, and state its exact value. [2 marks]
(iii) The $x$-coordinate of $P$ at time $t$ is $x_P = \cos\!\left(\dfrac{\pi}{3}t\right)$. Using the result from part (a), find all values of $t$ in $0 \leq t \leq 6$ for which $$x_P(t+1) \cdot x_P(t-1) = -\frac{1}{2},$$ and interpret geometrically what this condition means for the position of $P$ on the unit circle. [5 marks]

Using the result from part (a), find the exact values of $PR$ and the circumradius $R$ of triangle $PQR$. [3 marks]

A colleague suggests that the surveyor could instead have measured $\angle PQR = 45^\circ$, with $PQ = 120\,\text{m}$ and $QR = 60\sqrt{6}\,\text{m}$ unchanged. Determine the number of distinct triangles $PQR$ satisfying these three measurements, and justify your answer. [2 marks]

Using the Law of Sines, find $AL$ in exact form. Hence show that $$d = \frac{500\sin^2 50°}{\sin 80°}\,\text{m.}$$ [4 marks]

A third observation point $C$ is placed on the coastline between $A$ and $B$ such that $AC = 200\,\text{m}$. The bearing of $L$ from $C$ is $\theta$, where $0° < \theta < 90°$. (i) Show that $\tan\theta = \dfrac{d}{CL_x}$, where $CL_x$ is the horizontal distance from $C$ to the foot of the perpendicular from $L$ to $AB$. Express $CL_x$ in terms of known lengths. [1]
(ii) Hence find the exact value of $\tan\theta$, and determine whether $\theta > 40°$ or $\theta < 40°$. Justify your answer. [2 marks]