Geometry and Trigonometry — Free Maths AA SL Practice Questions

Find the coordinates of the point where $L$ crosses the $x$-axis. [2 marks]

A second line $M$ has equation $y = 2x - 4$. Determine the coordinates of the point of intersection of $L$ and $M$. [3 marks]

Find the coordinates of the point $P$ where line $L$ intersects the line $y = x$. [2 marks]

A second line $M$ passes through $P$ and is perpendicular to $L$. Determine whether $M$ passes through the point $C(3,\,4)$. [3 marks] Total: [8 marks]

The line found in part (a) meets $BC$ at point $D$. Determine the coordinates of $D$. [2 marks]

Show that $D$ divides $BC$ in the ratio $BD:DC = 4:1$, and hence determine the ratio of the area of triangle $ABD$ to the area of triangle $ABC$. [2 marks]

Find the equation of $L$ in the form $y = mx + c$. [1 mark]

A second line $M$ has equation $y = x - 2$. Determine the coordinates of the point of intersection of $L$ and $M$. [2 marks]

Line $M$ is perpendicular to line $L$ and passes through the point $Q(6, 3)$. Find the equation of line $M$ in the form $y = mx + c$. [2 marks]

Find the coordinates of the point of intersection of lines $L$ and $M$. [3 marks]

Calculate the length of the ladder $l$. [2 marks]

A second ladder of length $5\,\text{m}$ leans against the same wall with its base also $2\,\text{m}$ from the wall. Determine the height reached by the second ladder and evaluate whether it reaches more than $1\,\text{m}$ higher up the wall than the first ladder. [4 marks]

Hence find the straight-line distance from $P$ to the top of the pole. [3 marks]

Given that the wheel completes one full revolution every $12$ seconds, find the value of $c$. [2 marks]

Find the exact values of $t$ in the interval $0 \leq t \leq 24$ for which $h(t) = 8.5$ metres, and justify that no other solutions exist in this interval. The following identities may be useful: $\cos(\theta) = \cos(-\theta)$, $\quad \cos(\theta) = \cos(\theta + 2\pi k)$ for $k \in \mathbb{Z}$ [4 marks]

Show that $QR = 12\,\text{cm}$. [2 marks]

Hence find the exact value of $\sin 2\angle RPQ$. [2 marks]

Write down the exact value of $\sin\theta$. [1 mark]

Hence find the exact value of $\sin 2\theta$. [2 marks]

Find the height of the tower, correct to 3 significant figures. [3 marks]

Hence find the length of $QR$, giving your answer correct to 3 significant figures. [2 marks]

A path from $A$ meets $BC$ at point $D$ such that $AD$ bisects angle $B\hat{A}C$. State the angle bisector theorem and hence determine the exact ratio $BD : DC$. [2 marks]

Find the length $BD$, giving your answer correct to 3 significant figures. [2 marks]

The surveyor claims that point $D$ lies closer to $B$ than to the midpoint $M$ of $BC$. Determine whether this claim is correct, justifying your answer with calculations. [1 mark]

Find the smallest angle of the triangle, correct to the nearest degree. [2 marks]

A fence is to be built along all three sides of the garden. Fencing costs USD 35 per metre. Determine the total cost of the fence, giving your answer correct to the nearest dollar. [1 mark]

Hence calculate the distance $AB$, giving your answer to the nearest metre. [2 marks]

The surveyor's measuring tape has a maximum length of $35\,\text{m}$. She claims she could verify $AB$ directly by repositioning $C$ so that $\angle BAC = 90°$, keeping $B$ fixed, and measuring a new baseline $AC'$ instead. Determine the minimum length of $AC'$ required for this method, and state whether tape is long enough. [3 marks]

Determine the equation of line $AB$ in the form $y = mx + c$. [2 marks]

The line $AB$ intersects the $y$-axis at point $C$. The midpoint of $AB$ is $M$. A second line passes through $M$ and is perpendicular to $AB$. Determine the $x$-coordinate of the point where this second line intersects the $x$-axis. [2 marks]

A third town $C$ lies on the road between $A$ and $B$ such that $AC : CB = 1 : 2$. Determine the coordinates of $C$. [2 marks]

A fourth town $D$ lies on the road such that $B$ is the midpoint of $AD$. Determine the coordinates of $D$ and hence find the distance $AD$. [3 marks] Constrained 6-mark version (faithful to original scope): A straight road passes through two towns. Town $A$ is at $(-2,\,1)$ and town $B$ is at $(4,\,7)$. (a) Show that the equation of the road is $y = x + 3$. [2 marks] (b) A third town $C$ lies on the road between $A$ and $B$ such that $AC : CB = 1 : 2$. Determine the coordinates of $C$. [2 marks] (c) A fourth town $D$ lies on the road such that $B$ is the midpoint of $AD$. Determine the coordinates of $D$. [2 marks]

Determine the equation of the line that represents the drone's flight path, giving your answer in the form $y = mx + c$. [2 marks]

The drone continues along the same path at the same constant speed. Determine the time at which the drone crosses the $x$-axis. [2 marks]

Determine the equation of line $L$ in the form $ax + by + d = 0$, where $a, b, d \in \mathbb{Z}$. [2 marks]

Line $M$ is perpendicular to $L$ and passes through the midpoint of $CD$. Lines $L$ and $M$ intersect at point $P$. Determine the coordinates of $P$ and hence find the distance $CP$. [4 marks]

Line $M$ is perpendicular to line $L$ and passes through the point $B(6,\,1)$. Find the $y$-intercept of line $M$. [2 marks]

Lines $L$ and $M$ intersect at point $C$. Determine the coordinates of $C$, and hence determine whether $C$ is closer to $A$ or to $B$. [4 marks]

Given that $x = 1.5$, calculate the value of $\theta$, giving your answer in degrees correct to 3 significant figures. [2 marks]

Find the value of $\theta$, where $0^\circ < \theta < 90^\circ$, for which $h + x$ is a maximum. [2 marks]

A second point $Q$ lies on the same horizontal ground with $PQ = 5\,\text{m}$. The angle of elevation of the top of the flagpole from $Q$ is $25°$. (i) Find the distance from $Q$ to the base of the flagpole, correct to $3$ significant figures. [1]
(ii) Using your results from (a) and (b)(i), determine the angle $\angle PBQ$, where $B$ is the base of the flagpole. Hence determine whether the flagpole lies between $P$ and $Q$, or outside segment $PQ$. Justify your answer. [3 marks]

At that point, the ship changes course and sails directly towards the lighthouse at a speed of $12\,\text{km h}^{-1}$. The ship must arrive at the lighthouse no later than 45 minutes after it changed course. Determine whether the ship arrives in time, justifying your answer with a calculation. [3 marks]

Calculate the height of the passenger after $25$ seconds. Give your answer correct to $3$ significant figures. [2 marks]

Find the first time $t > 0$ at which the passenger is exactly $32$ metres above the ground. Give your answer correct to the nearest second. [3 marks]

A second tree, $S$, stands on the opposite bank. From point $A$, the bearing of $S$ is $038°$. From point $C$, the bearing of $S$ is $018°$. Determine the distance $AS$. Give your answer correct to $3$ significant figures. [3 marks]

Determine the shortest distance from $A$ to the line $BC$. [3 marks]

Calculate the distance $LC$. [2 marks]

Hence find the shortest distance from $B$ to the line $LC$. The following formula is provided: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ [2 marks]

Determine the area of the garden. [2 marks]

A straight path is to be laid from vertex $A$ to the fence $BC$, perpendicular to $BC$. Calculate the length of this path. [2 marks]

Two students calculate the height $h$ of the balloon. - Student A uses $h = PR\tan 38°$. - Student B uses $h = QR\tan 52°$. Calculate $h$ using both expressions and explain why the two values confirm the answer is correct. Booklet formulae available: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}, \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$ [3 marks]

Calculate the area of triangle $ABC$. [3 marks]