Exercises: Triangles

To decorate her room, Lucía has cut out a fabric pennant shaped like a triangle. On measuring its sides, she found that they measure $15\text{ cm}$, $15\text{ cm}$ and $20\text{ cm}$. How is this triangle classified according to the length of its sides?

In a triangle, we know the measure of two of its interior angles: $\hat{A} = 45^\circ$ and $\hat{B} = 75^\circ$. Calculate the measure of the third angle $\hat{C}$ knowing that the sum of the angles in any triangle is always $180^\circ$.

Marcos has three wooden rods to build a triangular structure. The lengths of the rods are $12\text{ cm}$, $5\text{ cm}$ and $4\text{ cm}$. The figure shows the three rods drawn to the same scale.

Explain with reasoning whether Marcos will be able to form a triangle with them or whether it will be impossible.

Classify the following triangles according to the measure of their angles (acute-angled, right-angled or obtuse-angled):

• Triangle 1: Its angles measure $90^\circ$, $40^\circ$ and $50^\circ$.

• Triangle 2: Its angles measure $110^\circ$, $30^\circ$ and $40^\circ$.

• Triangle 3: Its angles measure $60^\circ$, $60^\circ$ and $60^\circ$.

A garden is shaped like an equilateral triangle. If one of its sides measures $8\text{ metres}$, how many metres of fencing will we need to enclose it completely? Calculate its perimeter.

An isosceles triangle has an unequal angle measuring $40^\circ$. If the other two angles are equal to each other, what is the measure of each of those equal angles?

Look at the "Give way" road sign. It is shaped like an equilateral triangle. State the measure of each of its interior angles without using a protractor. Justify your answer based on the properties of this type of polygon.

In a right-angled triangle, the two sides that form the $90^\circ$ angle measure $6\text{ cm}$ and $8\text{ cm}$. What are these two sides called in the mathematical terminology of right-angled triangles? And what is the longest side opposite the right angle called?

Marcos wants to build the frame for a triangular mirror for his bedroom and has cut three wooden slats measuring $4\text{ cm}$, $5\text{ cm}$, and $10\text{ cm}$. When showing them to his sister, he states: "Since the three slats have different lengths, I'll be able to form a scalene triangle without any problem."\na) Find the error in Marcos's statement and explain why it is physically impossible to build the frame with those measurements, applying the triangle inequality.\nb) If Marcos decides to replace only the $10\text{ cm}$ slat with a new one, what is the maximum integer length (in centimetres) that the new slat can have to be able to close the triangle together with the $4\text{ cm}$ and $5\text{ cm}$ ones?\nc) If he wanted to form an isosceles triangle reusing the $4\text{ cm}$ and $5\text{ cm}$ slats, state what two different integer measurements the replacement slat could have.

In Art class, Lucía is designing a stained-glass window with triangular glass pieces. In one of the pieces, she has measured two of its interior angles with a protractor, which turn out to be $27^\circ$ and $38^\circ$.\na) Before calculating the exact value of the third angle, make a reasoned estimation of whether this angle will be acute, right, or obtuse (observe whether the sum of the two known angles exceeds $90^\circ$ or not). Based on your estimation, indicate what type of triangle you expect to obtain according to its angles.\nb) Calculate the exact value of the third angle using the property of the sum of the interior angles of a triangle and check whether your estimation was correct.\nc) Knowing that the three sides of this piece have different lengths, classify the triangle according to its sides.

In a graphic design studio, they are creating a new logo shaped like a triangle. We know that two of its interior angles measure $42^\circ$ and $68^\circ$.

1. a) Calculate the measure of the third angle of the logo.
2. b) Classify the triangle by its angles (acute-angled, right-angled or obtuse-angled).
3. c) Classify the triangle by its sides (equilateral, isosceles or scalene).

A carpenter has three wooden battens to make a triangular frame for a mirror. The battens measure $15\text{ cm}$, $8\text{ cm}$ and $24\text{ cm}$.

1. a) Explain, using the triangle inequality property, whether it is possible to construct the frame with those measurements.
2. b) If he decides to replace the batten of $24\text{ cm}$ with one of $20\text{ cm}$, could he construct it now? Explain your answer.

A tent has a front shaped like an isosceles triangle. The base of the tent measures $2,4\text{ metres}$ and the two equal sides measure $2\text{ metres}$ each.

1. a) Use Pythagoras' theorem to calculate the height of the tent.
2. b) Calculate the area of canvas needed to cover only the front of the tent.

Three villages, Altea ($A$), Belmonte ($B$) and Calar ($C$), are located so that they form a scalene triangle, as shown in the figure. A telecommunications tower is to be installed exactly the same distance from all three villages.

1. a) Which notable point of the triangle represents the location of the tower?

1. b) Which notable lines did you have to draw to find that point? Draw a rough sketch of the situation.

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

A children's playground has a sandpit shaped like a right-angled triangle. We know that the longest side (the hypotenuse) measures $13\text{ metres}$ and one of the sides forming the right angle measures $5\text{ metres}$.

1. a) Calculate the length of the other side of the sandpit.
2. b) Find the total perimeter of the sandpit.
3. c) Calculate the surface area occupied by the sand.

A triangular road sign has been designed in the shape of an equilateral triangle with a side length of $60\text{ cm}$.

1. a) If we want to put a reflective border around the sign, how many metres of reflective material will we need?
2. b) If we draw the three angle bisectors of the triangle, what is the point where they cross called? What special property does this point have with respect to the sides of the sign?

Look at the following figure composed of two triangles. Triangle $ABC$ is right-angled at $B$ and triangle $ACD$ is isosceles with $\overline{AC} = \overline{AD}$.

If $\overline{AB} = 8\text{ cm}$ and $\overline{BC} = 6\text{ cm}$, calculate:

1. a) The length of side $\overline{AC}$.
2. b) The total perimeter of the shaded figure $ABCD$ knowing that $\overline{CD} = 7\text{ cm}$.

In a garden, a fence is to be put up to enclose a flower bed in the shape of an isosceles triangle. The unequal angle measures $100^\circ$.

1. a) What is the measure of each of the other two angles of the flower bed?
2. b) If we draw the median from the vertex of the $100^\circ$ angle to the opposite side, into what two types of triangles is the region divided? Classify them by their angles.

A graphic design team must create a logo containing a triangular figure strictly fulfilling three geometric conditions:\n1. The triangle must be isosceles and also obtuse-angled.\n2. The obtuse angle is the unequal angle and its value in degrees must be a multiple of $10^\circ$.\n3. Each of the two equal angles must measure strictly more than $25^\circ$.\na) Set up the mathematical relationship between the three interior angles and deduce step by step all possible combinations of angles (triplets) that satisfy these three conditions.\nb) Could any valid solution exist if we added the condition that the equal angles measure more than $45^\circ$? Justify your answer using the definition of an obtuse-angled triangle.

On a road map of Castilla y León, three provincial capitals form a triangle: Palencia, Valladolid, and Zamora. The distances measured in a straight line are approximately:\n- Between Palencia and Valladolid: $48\text{ km}$.\n- Between Valladolid and Zamora: $96\text{ km}$.\na) The Junta de Castilla y León is studying the feasibility of opening a direct road between Palencia and Zamora. Applying the triangle inequality, calculate between what minimum and maximum values (excluding the endpoints) the straight-line distance between Palencia and Zamora must lie so that the route is not flattened into a single straight line.\nb) If the actual straight-line distance between Palencia and Zamora turned out to be $110\text{ km}$, check using the Pythagorean theorem whether these three cities could form a right-angled triangle with a right angle at Valladolid (where the Palencia-Zamora section would be the hypotenuse).\nc) If a tourist brochure mistakenly indicated that the direct distance between Palencia and Zamora is only $40\text{ km}$, explain why that figure is geometrically impossible without the need to measure on the ground.

In a video game design workshop, a programmer needs to create a character shaped like an isosceles triangle. The top vertex angle measures $20^\circ$ less than three times one of the base angles. Calculate the measure of the three angles of the character so that the design is geometrically correct.

Three mountain villages, Alpedrete, Becerril and Cercedilla, want to build a telephone tower that is exactly the same distance from all three so that everyone has the same coverage.

1. a) What is the name of the notable point of the triangle where the tower should be placed?

1. b) If the triangle formed by the villages is obtuse, explain with reasoning whether the tower will be inside or outside the area enclosed by the three villages.

1. c) Draw a rough sketch of the situation: the obtuse triangle with the three villages, the lines you use to find the point, and the position of the tower.

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

An architect is designing a ramp for a skatepark. He has calculated that the main support must form a triangle with sides measuring $x + 2$ metres, $2x - 1$ metres, and $10$ metres.

1. If the triangle must be isosceles, with sides $x + 2$ and $2x - 1$ being equal, what must the value of $x$ be?

2. If $x = 3$, is it possible to construct that triangle? Justify your answer using the triangle inequality theorem.

In the logo of a new sportswear brand, two rectangles intersect forming several triangles. In the central triangle $ABC$, we know that the angle at $A$ measures $45^\circ$ and that the exterior angle at vertex $B$ measures $115^\circ$.

1. Calculate the three interior angles of triangle $ABC$.

2. Classify the triangle by its sides and by its angles.

A garden is shaped like a rectangle of $12 \times 8$ metres. To install an irrigation system, the garden has been divided by drawing a diagonal. In one of the two resulting triangles, a circular sprinkler is to be placed at the incentre.

1. Explain what angle bisectors are and why placing the sprinkler at the incentre is the best option for it to be equidistant from the three sides of the triangle (the two edges of the garden and the diagonal).

2. Draw a sketch of the rectangle, the diagonal, and the incentre of one of the triangles.

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

Observe the attached figure, which represents a metal piece. It consists of a large right-angled triangle and a smaller one inside it. We know that the horizontal leg of the large triangle is $15 \text{ cm}$ and its area is $60 \text{ cm}^2$. The small triangle is similar and has a base of $5 \text{ cm}$.

1. Calculate the height of the large triangle.

2. Using the logic of proportions, calculate the area of the small triangle. (Hint: if the base is one-third, what happens to the height?).

An explorer has placed a rope of $24$ metres around a triangular plot of land to set up his camp.

1. If the plot is an isosceles triangle and one of the equal sides measures $9 \text{ metres}$, how long is the remaining side?

2. If the explorer wanted the triangle to be right-angled with sides of $6 \text{ m}$, $8 \text{ m}$ and $10 \text{ m}$, would the same rope work? Check also if Pythagoras' theorem holds in this latter case.

In an advanced geometry exam, a triangle is shown where the three medians intersect at a point $G$ (Centroid). If the median coming from vertex $A$ has a total length of $12 \text{ cm}$, calculate the length of the two segments into which point $G$ divides this median. (Remember the property: the centroid divides the median into two parts, one being twice the length of the other).

To find out whether three rigid rods of lengths $a = 8\text{ cm}$, $b = 15\text{ cm}$, and $c = 25\text{ cm}$ can be joined at their ends to build a triangular structure, two methods are proposed in the classroom:\n- Method 1 (Complete check): Set up and check the three triangle inequalities one by one: $a < b + c$, $b < a + c$, and $c < a + b$.\n- Method 2 (Reduced check): Locate the longest rod ($c$) and check only whether that rod is less than the sum of the other two ($c < a + b$).\na) Apply both methods to the case of the rods of $8\text{ cm}$, $15\text{ cm}$, and $25\text{ cm}$ and write down the conclusion reached in each one.\nb) Mathematically justify why Method 2 is always sufficient: why, if the longest side is less than the sum of the other two, can we be absolutely certain that the other two inequalities will always hold without needing to calculate them?\nc) If we decide to remove the $25\text{ cm}$ rod and replace it with a new rod of integer length $x$ to form a triangle together with the $8\text{ cm}$ and $15\text{ cm}$ ones:\n1) Calculate what exact integer length $x$ must have for the triangle to be right-angled with that rod acting as the hypotenuse, verifying it with the Pythagorean theorem.\n2) State what integer values $x$ could take for the resulting triangle to be isosceles and geometrically possible.

During a high-mountain trek in the Picos de Europa National Park, three shelters ($A$, $B$, and $C$) are connected by straight trails forming a triangle. The mountain guide checks the fact sheets:

• The angle at shelter $A$ is three times the angle at shelter $B$.

• The angle at shelter $C$ measures exactly $20^\circ$.

• The straight-line distance between shelter $A$ and shelter $B$ is $7\text{ km}$.

• The straight-line distance between shelter $B$ and shelter $C$ is $18\text{ km}$.

1. a) Model the situation by setting up an equation with the interior angles of the triangle and find the exact measure of the angles at shelter $A$ and shelter $B$.
2. b) Classify the triangle formed by the three shelters both by its angles (acute, right, or obtuse) and by its sides (equilateral, isosceles, or scalene).
3. c) A hiker claims: "To go directly from shelter $C$ to shelter $A$ in a straight line, we will have to walk more than $25\text{ km}$". Prove mathematically, using the triangle inequality, why the hiker's claim is impossible.
4. d) The 112 emergency service needs to install an emergency antenna that is at the exact same distance from all three shelters. Name the notable point (triangle center) that the technicians must locate and explain which lines they must draw on the map to find it.