Exercises: Solving non-right-angled triangles

In a triangle $ABC$, the sides $a = 5$ cm and $b = 8$ cm are known, as well as the included angle $C = 60^\circ$ between them. Calculate the length of side $c$.

The three sides of a triangle measure $a = 3$ m, $b = 5$ m and $c = 7$ m. Determine the size of the largest angle in this triangle.

In a triangle $ABC$, the angles $A$ and $B$ measure $30^\circ$ and $105^\circ$ respectively. If the side opposite angle $A$ has a length of $a = 8$ cm, find the length of side $c$ (opposite angle $C$).

In a triangle, the following data are known: $a = 12$ m, $b = 8$ m and angle $A = 120^\circ$. Calculate the size of angle $B$.

Find the area of a triangle $ABC$ for which the sides $b = 14$ cm, $c = 20$ cm and the angle between them, $A = 60^\circ$, are known.

The adjacent sides of a parallelogram measure $6$ cm and $10$ cm, forming an obtuse angle of $120^\circ$. Calculate the length of the longer diagonal of the parallelogram.

In an isosceles triangle, the non-equal side (the base) measures $12$ m and the equal angles measure $75^\circ$ each. Determine the length of the equal sides using the sine rule.

In a triangle $ABC$, side $a$ measures $10$ cm and the opposite angle $A$ measures $45^\circ$. Calculate the radius $R$ of the circumscribed circle of the triangle.

Seán has to calculate the length of side $a$ of a triangle for which he knows $b = 5\text{ cm}$, $c = 8\text{ cm}$, and the included angle $A = 120^\circ$. In his notebook, he writes the following resolution:

1) Point out the calculation error Seán made when working with the signs.

2) Without having to redo the calculations yet, explain why a result of $a = 7\text{ cm}$ is geometrically impossible in this triangle (relate the magnitude of angle $A$ to the relative measures of the sides).

3) Correct the procedure and calculate the correct value of side $a$ rounded to two decimal places.

In a triangle, we know sides $a = 6\text{ cm}$ and $b = 6\text{ cm}$, and the included angle between them $C = 62^\circ$.

1) Estimate: before performing any numerical calculations with formulas, reason whether the third side $c$ will be slightly greater than, less than, or equal to $6\text{ cm}$. To justify this, imagine what the figure would look like if the angle were exactly $60^\circ$ (equilateral triangle) and analyze what happens when opening the angle two degrees more.

2) Check: calculate the exact length of side $c$ using the law of cosines (round to two decimal places).

3) Assess whether the obtained result confirms your qualitative estimate and calculate the absolute error you would have made if you had directly approximated $c \approx 6\text{ cm}$.

Two ships leave a port simultaneously along straight routes that form an angle of $75^\circ$ with each other. The first ship sails at a constant speed of $15$ knots and the second at $20$ knots. Calculate the distance in nautical miles separating the two ships after $2$ hours of navigation.

To measure the height $h$ of a wind turbine, we stand at a point $A$ on the ground and measure the angle of elevation to the top point, obtaining $40^\circ$. We move $30$ metres in a straight line towards the base of the wind turbine to reach a point $B$, from which the new angle of elevation is $55^\circ$. Calculate the height of the wind turbine.

In the design of an articulated robotic arm, the first link measures $40$ cm and the second measures $35$ cm. If the angle formed by the first link with the horizontal base is $50^\circ$ and the end of the second link touches the horizontal base exactly, calculate the two possible distances from the origin of the arm to the contact point on the base.

A piece of farmland has the shape of a quadrilateral $ABCD$. The lengths of its sides are $AB = 50$ m, $BC = 40$ m, $CD = 30$ m and $DA = 60$ m. Knowing that the angle at vertex $B$ is $70^\circ$, calculate the total area of the land by dividing it into two triangles.

Two tugboats pull a barge using two cables that form an angle of $45^\circ$ with each other. If the tensions exerted by the cables are $4500$ N and $6000$ N, find the magnitude of the resultant force and the angle this resultant forms with the cable exerting the force of $6000$ N.

A surveillance drone is at a point $C$ observing a straight stretch of road bounded by two cities $A$ and $B$ that are $5$ km apart. The angles of depression from the drone to the cities are such that in the triangle formed, the angle at $A$ is $35^\circ$ and the angle at $B$ is $62^\circ$. Calculate the altitude at which the drone is flying above the road.

A regular heptagon (a polygon with $7$ equal sides) is inscribed in a circle of radius $10$ cm. Using the cosine rule and/or sine rule in the central triangles, calculate the perimeter and area of the heptagon.

A solar panel $2$ metres long is to be installed on a pitched roof that has an inclination of $15^\circ$ to the horizontal. To maximize its performance, the panel must form an angle of $35^\circ$ with respect to the horizontal. Calculate the length $x$ of the vertical support that must be placed at the upper end of the panel to connect it to the roof.

The three sides of a triangle are known: $a = 4\text{ cm}$, $b = 7\text{ cm}$, and $c = 10\text{ cm}$. To solve it completely, you must find its three angles. Two classmates propose different paths to calculate the second angle after having obtained the first:

• Aoife: "First, I calculate the smallest angle, $A$, with the law of cosines. Then, to make the calculations faster and simpler, I calculate the largest angle, $C$, using the law of sines ($\frac{a}{\sin A} = \frac{c}{\sin C}$), and finally $B = 180^\circ - A - C$".

• Seán: "It is dangerous to calculate $C$ using the law of sines. It is much safer to calculate $C$ directly with the law of cosines, or use the law of sines to calculate $B$, but never for $C$".

1) Mathematically justify Seán's objection: what limitation does the $\arcsin$ function on the calculator have when looking for angles that could be obtuse?

2) Check the flaw in Aoife's method: calculate angle $A$ using the law of cosines and show what incorrect value for $C$ the law of sines would yield if applied directly.

3) Apply the safe path proposed by Seán to calculate the three angles of the triangle accurately.

During the fire prevention campaign of the Department of Agriculture, Food and the Marine (Coillte forest fire prevention programme), two fire lookout towers, Slieve Bloom Tower ($A$) and Mount Leinster Tower ($B$), located at a straight-line distance of $12\text{ km}$ from each other, detect a column of smoke at a point $C$. The observers take references relative to the straight segment $AB$: from tower $A$, the smoke is observed at an angle of $48^\circ$, and from tower $B$, at an angle of $74^\circ$ (both lines of sight directed toward the same side of line $AB$).

1) Model the situation geometrically and calculate the straight-line distance from each of the two towers to the fire source $C$.

2) To take action on the ground, two options are evaluated:

• A water-bombing helicopter based at tower $A$, which takes off immediately and flies in a straight line toward the fire at a constant speed of $190\text{ km/h}$.

• A fire engine off-road vehicle based at tower $B$, which travels at an average speed of $45\text{ km/h}$ along forest tracks, covering an actual route estimated to be $25\,\%$ longer than the straight-line distance.

Calculate the travel time (in minutes and seconds) for each resource and determine with reasoning which one will reach the fire source first.

The sides of a triangle are given by the expressions $x$, $x+2$ and $x+4$ centimetres. Knowing that the largest angle of the triangle measures $120^\circ$, set up the corresponding equation, find the exact value of $x$ and calculate the perimeter of the triangle.

In a triangle $ABC$, it is known that side $a = 10$ cm, angle $B = 30^\circ$ and that the total area of the triangle is $20\sqrt{3}$ cm$^2$. Set up a system of equations to find the exact length of sides $b$ and $c$, and find the size of angles $A$ and $C$.

Prove geometrically or using trigonometric rules (sine and cosine rules) that in any triangle $ABC$ the projection formula holds: $a = b \cos C + c \cos B$. Next, use this identity together with the sine rule to prove the sine compound angle formula: $\sin(B+C) = \sin B \cos C + \cos B \sin C$.

The hands of a wall clock measure $12$ cm (the minute hand) and $8$ cm (the hour hand). Calculate the exact distance in centimetres separating the tips of both hands when the clock shows exactly 10:10. Take into account the proportional displacement of the hour hand.

Two aeroplanes fly at the same altitude. Aeroplane 1 is at point $A$ and flies North at a constant speed of $800$ km/h. Aeroplane 2 is at point $B$, located exactly $200$ km North-East ($45^\circ$ from North) of point $A$. If aeroplane 2 flies West at a constant speed $v$ and both collide at point $C$, determine the value of speed $v$ and the time it takes for them to collide.

In a triangle $ABC$ with sides $a=13$ cm, $b=14$ cm and $c=15$ cm, the circumscribed circle of radius $R$ and the inscribed circle of radius $r$ are drawn. Calculate the area of the triangle using Heron's formula and, from it, obtain the exact values of $R$ and $r$.

Given the triangle $ABC$ with sides $a = 7$ cm, $b = 5$ cm and $c = 8$ cm, the internal angle bisector of angle $C$ is drawn, intersecting side $AB$ at point $D$. Applying the cosine rule to find angle $C$ and using the properties of areas or the sine rule, calculate the exact length of the angle bisector segment $CD$ (denoted as $x$).

You can do it on paper and send a photo with your answer.

A regular square-based pyramid has a base side length of $100$ m. Knowing that the dihedral angle formed by two adjacent triangular faces is $105^\circ$, set up the geometry of the appropriate cross-section and calculate the total height of the pyramid.

A land surveyor needs to reconstruct the original measurements of a triangular plot $ABC$ from an incomplete old report that only includes three exact data points:

• The area of the plot is $80\text{ m}^2$.

• Boundary $b$ is exactly twice as long as boundary $a$ ($b = 2a$).

• The included angle $C$ satisfies $\cos C = 0.6$.

1) Using the fundamental trigonometric identity and the formula for the area of a triangle in terms of two sides and the included angle, set up an equation and deduce the exact lengths of sides $a$ and $b$.

2) Calculate the length of the third side $c$ by applying the law of cosines.

3) Inverse problem: explain why there is another plot non-congruent with the previous one that has the exact same area ($80\text{ m}^2$) and the same lengths $a$ and $b$, but a different shape. Find the value of its new included angle $C'$, calculate the length side $c$ would have in this second configuration, and determine which of the two plots would require fewer meters of fencing for its perimeter enclosure.

In Math class, Aoife solves an exercise about delimiting a triangular plot $ABC$ with known measurements $a = 14\text{ m}$, $b = 20\text{ m}$, and angle $A = 35^\circ$. She presents the following work on the blackboard:

"I apply the law of sines:

Since the three angles add up to $180^\circ$, we have $C = 180^\circ - 35^\circ - 55^\circ = 90^\circ$. Since the triangle turns out to be a right triangle at $C$, I calculate the remaining side $c$ using the Pythagorean theorem: $c = \sqrt{20^2 - 14^2} = \sqrt{204} \approx 14.28\text{ m}$".

1) Identify and explain in detail the two conceptual errors made by Aoife:

a) Why is her deduction for angle $B$ incomplete and omitting a valid geometric solution?

b) Even assuming that angle $C$ were $90^\circ$, why is her application of the Pythagorean theorem incorrect?

2) Solve the problem with full rigor: determine the two possible triangles that satisfy the initial data, indicating for each of them the values of angles $B$ and $C$ and the length of side $c$.

3) If fencing the perimeter of the plot costs $45\text{ €/m}$, calculate the cost difference between choosing one configuration or the other.

Topic challenge. Redesign your municipality's environmental monitoring network using the official viewer of the National Geographic Institute (IGN, Iberpix).

1. Data collection and counting: Select 4 high ground points in your area. Apply counting techniques ($C_{4,3}$) to calculate how many different coverage triangles you could form. Choose one of them, record its coordinates or distances, and express its angles in both sexagesimal degrees and radians.

2. Oblique triangle solving: Determine all unknown sides and angles of this triangle by accurately applying the law of sines and the law of cosines in GeoGebra or a spreadsheet.

3. Height calculation and compound ratios: To install an antenna on one of the vertices, set up its calculation by solving right triangles with staggered lines of sight (for example, at distances $d$ and $2d$, or with angles $\alpha$ and $2\alpha$). Use the double-angle or angle subtraction formulas to solve for the tower's height without measuring it directly.

4. Reasoned proposal: Prepare a brief technical report with a map screenshot, the analytical development of the trigonometric ratios used, and the coverage conclusions.

Assessment criteria:

• Rigour in converting degrees to radians and in using identities (sum/difference or double angle).

• Correct algebraic and geometric solving of right and oblique triangles using the law of sines and the law of cosines.

• Appropriate application of combinatorial counting and accuracy with real cartographic data.