Triangles

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 $180^\circ$.

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?

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?

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 $60^\circ$.

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?

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).

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.

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 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.

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?).

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.

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.