Geometry and Trigonometry

A coordinate $A(2, 3)$ is reflected in the $y$-axis to point $B$. What are the coordinates of $B$?

Which of the following lines is perpendicular to the line $y = 4x - 2$?

A line has the equation $2x + y = 6$. At what point does this line cross the $x$-axis?

The side of a square building is represented by the line segment from $(1, 1)$ to $(1, 5)$ on a site plan. What is the length of this side?

The diagram shows the cross-section of a sloping roof represented by the line $L_1$. The equation of $L_1$ is $2x + 5y - 20 = 0$. A support beam $L_2$ is perpendicular to the roof and passes through the point $(8, 6)$. Find the $y$-intercept of the line representing the support beam $L_2$.

A new pipeline is being laid in a straight line between two maintenance stations $S_1(4, -2)$ and $S_2(16, 8)$. An inspection hatch $H$ is located on the pipeline such that it divides the distance from $S_1$ to $S_2$ in the ratio $1:3$. Find the coordinates of the hatch $H$.

A land surveyor is mapping a triangular park. The coordinates of the vertices of the park are $A(2, 3)$, $B(10, 3)$, and $C(6, 9)$, where the units are in kilometers. A straight walking path is to be constructed that starts at the midpoint of $AC$ and ends at the midpoint of $BC$. Find the length of this walking path.

A triangular plot of land has vertices $U(1, 2)$, $V(7, 2)$, and $W(1, 10)$. A surveyor needs to find the coordinates of the circumcenter (the point equidistant from all vertices). Find the $y$-coordinate of the circumcenter.

A map shows a circular lake centered at $C(4, 5)$ with a radius of $5$ km. A straight road is represented by the line $x = 9$. At how many points does the road touch or cross the lake boundary?

Two shipping routes are defined by the equations $L_1: y = 0.75x + 2$ and $L_2: 3x + 4y = 44$. A lighthouse is located at the intersection of these two routes. Determine the distance of the lighthouse from the origin $(0,0)$.

On a scale drawing, a fence runs from point $(2, 2)$ to $(10, 2)$. A gate is installed at the midpoint. What are the coordinates of the gate?

Point $M(3, 5)$ is the midpoint of line segment $AB$. If $A$ is at $(1, 2)$, what are the coordinates of $B$?

Find the coordinates of the midpoint of the line segment with endpoints $(-5, 2)$ and $(5, 2)$.

A rectangle on a grid has vertices at $A(0,0), B(4,0), C(4,3)$ and $D(0,3)$. Calculate the length of the diagonal $AC$.

A circle is defined by the diameter with endpoints $A(-2, -1)$ and $B(4, 7)$. Find the coordinates of the center of the circle.

In a 3D simulation, a particle moves from $A(1, 2, 3)$ to $B(4, 6, 3)$ and then to $C(4, 6, 8)$. What is the total distance traveled by the particle?

A square has a diagonal with endpoints at $(1, 1)$ and $(5, 5)$. What is the length of one side of the square?

A drone path is defined by the segment connecting $K(-20, -10)$ and $L(20, 30)$. The drone is currently at the midpoint of $KL$. It needs to travel to a landing pad located at the origin $(0, 0)$. What is the distance the drone needs to travel to reach the landing pad?

A cable is stretched between two poles at $P_1(-4, 12)$ and $P_2(8, -4)$. A support bracket is placed at the midpoint $M$. A second cable connects $M$ to a third pole at $P_3(10, 10)$. Calculate the length of the second cable.

A robot moves in a factory from station $A(2, 3)$ to station $B(14, 11)$. It must stop at a charging point $P$ that is exactly $5$ units away from station $A$ along the line $AB$. Find the coordinates of the charging point $P$.

A wheelchair ramp is being designed to reach a doorway. The ramp follows the line $L$ shown in the diagram. If the ramp must have a constant gradient of $0.08$ and the doorway is at a height of $1.2$ meters above the ground ($y$-intercept is $0$), find the horizontal distance $x$ from the start of the ramp to the doorway.

A straight path is constructed on a hillside. The path starts at point $A(10, 25)$ and ends at point $B(50, 45)$, where coordinates represent horizontal distance and vertical height in meters respectively. Calculate the gradient (slope) of the path $AB$.

The gradient of the line passing through $(k, 4)$ and $(2, 10)$ is $3$. Find the value of $k$.

In the diagram, a ladder leans against a wall. The foot of the ladder is $1.8$ m from the wall and it reaches $4.5$ m up the wall. What is the gradient of the ladder relative to the ground?

The cross-section of a roof is modeled by two line segments. The left side passes through $(-4, 2)$ and $(0, 5)$. The right side is perpendicular to the left side and passes through the peak at $(0, 5)$. Determine the gradient of the right side of the roof.

The position of a particle at time $t$ is given by $s(t) = 0.5t^2 + 2t$. Find the gradient of the distance-time graph (the velocity) at $t = 3$.

Find the gradient of the line that passes through the midpoint of $A(2, 4)$ and $B(6, 12)$ and also passes through the origin.

The cross-section of a dam wall is modeled on a coordinate system. The front face of the dam follows the line segment from $(0, 100)$ to $(40, 0)$. A support brace is to be installed at a right angle to this face, connecting to the ground at point $(60, 0)$. Find the gradient of the support brace.

A kite's string is taut and follows the line $3x - 4y + 12 = 0$. A bird flies along a path perpendicular to the string, passing through the point $(0, 8)$. Find the equation of the bird's path.

A technician is calibrating a sensor that measures the temperature gradient in a cooling tower. The temperature $T$ ($^{\circ}C$) at height $h$ (meters) is given by a linear model. At $h=10$, $T=85$. At $h=50$, $T=45$. Calculate the rate of change of temperature with respect to height.

A taxi company charges a fixed booking fee plus a constant rate per kilometer traveled. The total cost $C$, in dollars, for a journey of $d$ kilometers is shown on the graph below. Find the equation that represents the relationship between $C$ and $d$.

A fence is modeled by the equation $y = mx + c$. The fence passes through $(0, -3)$ and has a gradient of $5$. Which equation represents the fence?

A cable is stretched between two poles. In a coordinate system, the cable is a straight line passing through $(1, 4)$ with a gradient of $0$. What is the equation of the line representing the cable?

An architect designs a roofline represented by the equation $3x + 4y = 24$. Find the coordinates where the roofline meets the $x$-axis.

Two straight fences are being installed in a field. Fence 1 follows the line $L_1$ with the equation $3x - 4y + 12 = 0$. Fence 2, $L_2$, is parallel to Fence 1 and passes through the point $(8, 2)$. Find the $y$-intercept of the line representing Fence 2.

A park ranger is mapping a straight hiking trail. On a coordinate map, the trail starts at point $A(2, 8)$ and ends at point $B(10, 2)$. A rest station is to be built at the midpoint of the trail. Determine the equation of the line that represents a service road passing through this rest station, perpendicular to the hiking trail $AB$.

A mountain slope can be modeled by the equation $y = 0.5x + 200$. A cable car station is located at $S(100, 300)$. Determine the shortest distance (perpendicular distance) from the station to the mountain slope.

A city planner is designing a new straight-line pipeline. On a map, the pipeline passes through the point $P(3, -2)$ and has a gradient of $m = \frac{3}{4}$. The planner needs to determine where the pipeline will intersect a boundary wall modeled by the line $2x + 3y = 12$. Find the $x$-coordinate of the intersection point between the pipeline and the boundary wall, rounding your answer to two decimal places.

A mountain biking trail is modeled as a straight line on a Cartesian map where the horizontal distance $x$ and vertical distance $y$ are measured in kilometers. The trail starts at a point $A(2, 5)$ and passes through a checkpoint at $B(6, 3)$. A new emergency access road is to be built that is perpendicular to the trail and passes through the midpoint of $AB$. Find the equation of this access road in the form $ax + by + d = 0$, where $a, b, d \in \mathbb{Z}$.

A telecommunications company is placing a signal tower at the intersection of two straight service boundaries. Boundary 1 is given by $x + y = 10$ and Boundary 2 is given by $2x - 3y = 5$. Determine the coordinates of the signal tower.

A sector has an arc length of $10$ cm and a radius of $5$ cm. Find the area of the sector.

A point $A$ on a circle has coordinates $(6, 8)$. The circle is centered at the origin $(0, 0)$. A second point $B$ is also on the circle. What is the distance between the center and point $B$?

The center of a circular Ferris wheel is located at $(0, 15)$ on a coordinate grid, where units are in meters. If the wheel has a diameter of $26$ meters, which of the following is the equation of the circle representing the Ferris wheel?

A circular stained-glass window is divided into $8$ equal sectors. If the radius of the window is $60$ cm, find the area of one sector in terms of $\pi$.

A circular clock face has a circumference of $94.2$ cm. What is the approximate area of the clock face? (Use $\pi \approx 3.14$)

The coordinates of the endpoints of the diameter of a circle are $A(1, 2)$ and $B(7, 10)$. Find the radius of the circle.

A car's windshield wiper of length $50$ cm sweeps through an angle of $135^\circ$. Find the area of the windshield cleaned by the wiper.

A belt drives two pulleys as shown in the diagram. The larger pulley has a radius of $15\text{ cm}$ and the smaller one has a radius of $8\text{ cm}$. If the larger pulley rotates through $120^{\circ}$, through what angle does the smaller pulley rotate?

A pizza with a diameter of $30\text{ cm}$ is cut into $8$ equal slices. If one slice is eaten, calculate the perimeter of the remaining pizza (the total boundary of the remaining shape), to one decimal place.

A square of side length $10\text{ cm}$ is inscribed inside a circle. Calculate the area of the region inside the circle but outside the square.

Find the length of the diagonal of a square with side length $10$ cm.

A ramp has a vertical rise of $1.2$ meters and a horizontal run of $10$ meters. Calculate the angle the ramp makes with the horizontal.

Two sides of a triangle are $5$ m and $12$ m. The angle between them is $90^{\circ}$. Find the length of the third side.

In the diagram, $O$ is the center of the circle. The points $A$ and $B$ lie on the circumference. If the area of the sector $AOB$ is $25\pi \text{ cm}^2$ and the radius is $10$ cm, find the angle $\angle AOB$ in degrees.

A surveyor is mapping a triangular park. The diagram shows the vertices of the park at points $A$, $B$, and $C$. The surveyor measures the internal angle at $B$ to be $42^{\circ}$ and the length of the path $BC$ to be $85$ m. The straight path $AB$ is $60$ m long. Calculate the size of the angle $\angle BAC$ to the nearest degree.

A pyramid has a square base of side $10$ cm and a vertical height of $12$ cm. Find the angle the slanted edge makes with the base diagonal.

The angle of depression from the top of a cliff $80$ m high to a boat at sea is $15^{\circ}$. How far is the boat from the base of the cliff?

A sculpture consists of a large metal ring of radius $2$ m. A support cable is attached from the top of the ring $T$ to a point $G$ on the ground, $5$ m from the center $C$ of the base of the ring. The ring is vertical. Calculate the angle the cable makes with the horizontal ground.

An observer at point $O$ on a cliff $80$ m above sea level sees two boats, $A$ and $B$. Boat $A$ is at a bearing of $030^{\circ}$ with an angle of depression of $15^{\circ}$. Boat $B$ is at a bearing of $120^{\circ}$ with an angle of depression of $10^{\circ}$. Calculate the distance between the two boats.

A piece of land is in the shape of a sector of a circle with radius $r$ and central angle $\theta$ radians. The perimeter of the land is $100$ m. Find the value of $\theta$ that maximizes the area of the land.

In triangle $XYZ$, $\angle Y = 90^\circ$, $XY = 15$ cm and $\angle X = 30^\circ$. Find the length of $YZ$.

The shadow of a vertical flagpole is 12 m long when the angle of elevation of the sun is $35^\circ$. Find the height of the flagpole.

In a right-angled triangle $PQR$, $PQ = 5$, $QR = 12$, and $\angle Q = 90^\circ$. Find $\sin(\angle PRQ)$.

A person looks down from a cliff 100 m high at a boat in the sea. The angle of depression is $15^\circ$. How far is the boat from the base of the cliff?

A pyramid has a square base of side 10 cm and a vertical height of 12 cm. Find the angle between a sloping face and the base.

In a right-angled triangle, the hypotenuse is 13 cm and the area is 30 cm$^2$. Find the length of the shorter leg.

A point $A$ is $10$ units from the origin $O$. The line $OA$ makes an angle of $210^\circ$ with the positive $x$-axis. Find the $y$-coordinate of point $A$.

A parcel of land is in the shape of a quadrilateral $ABCD$. $AB = 45$ m, $BC = 60$ m, $CD = 70$ m, $\angle ABC = 90^{\circ}$, and $\angle ADC = 55^{\circ}$. Calculate the length of the diagonal $AC$ and use it to find the area of the parcel of land to the nearest square meter.

A flagpole $FG$ stands on top of a building $GH$. From a point $P$ on the ground, the angle of elevation to the bottom of the pole $G$ is $35^{\circ}$ and the angle of elevation to the top of the pole $F$ is $48^{\circ}$. If the building is $20$ m tall, find the height of the flagpole $FG$.

A regular hexagon with side length $10$ cm is inscribed in a circle. A segment of the circle is formed by one side of the hexagon. Calculate the area of this single segment (the region between the chord and the arc) to one decimal place.

A surveyor is measuring the distance across a small lake between two points, $A$ and $B$. From a third point $C$, the surveyor finds that the distance $AC = 120$ m. The angle at $A$ ($\angle BAC$) is measured as $45^\circ$ and the angle at $B$ ($\angle ABC$) is measured as $70^\circ$, as shown in the diagram. Calculate the distance $AB$ across the lake, rounded to the nearest meter.

A triangle has side lengths 10, 15, and an unknown side $x$. If the angle opposite side 10 is $30^{\circ}$, what is the value of $\sin(X)$ where $X$ is the angle opposite side 15?

A survivor on an island $I$ is spotted by two rescue boats, $A$ and $B$. $A$ and $B$ are 10 km apart. Boat $A$ calculates $\angle IAB = 52^{\circ}$ and Boat $B$ calculates $\angle IBA = 76^{\circ}$. Find the distance from Boat $B$ to the survivor.

In triangle $JKL$, $\angle J = 25^{\circ}$, $\angle K = 35^{\circ}$, and the longest side is 20 cm. Find the length of the shortest side.

In a non-right-angled triangle $ABC$, $a = 10$, $b = 12$, and $\angle A = 45^{\circ}$. Find the number of possible triangles that can be formed with these dimensions.

A drone flies from point $P$ to point $Q$, a distance of 150 m. It then turns through an angle of $40^{\circ}$ and flies to point $R$. If the bearing of $R$ from $P$ is $075^{\circ}$ and $Q$ is due East of $P$, find the distance $QR$.

Given triangle $ABC$ with $c = 10$, $\angle A = 30^{\circ}$ and $\angle B = 100^{\circ}$, calculate the perimeter of the triangle to one decimal place.

A boat sails from port $O$ on a bearing of $060^{\circ}$ for $12$ km to reach point $A$. It then changes course and sails on a bearing of $140^{\circ}$ until it reaches point $B$, which is due East of $O$. Calculate the distance $AB$.

A telecommunications mast $M$ is supported by two cables $MA$ and $MB$ anchored to the ground. Cable $MA$ is $45$ m long and makes an angle of $55^{\circ}$ with the horizontal ground. Cable $MB$ is anchored on the opposite side of the mast and makes an angle of $40^{\circ}$ with the ground. Find the length of cable $MB$.