U4. flat Shapes. Similarity

Prior knowledge

Play this Kahoot to review some basics on Geometry

Check this web for prior knowledge required for this unit.

Also review some concepts about circles.

Polygons

If you would like to, try breathing in sync with this image (*):

(*) Source.

Polygon

A polygon is a flat, closed shape made of straight line segments.
These segments are called sides, and they meet at vertices.

Convex Polygon

A convex polygon is a polygon where all its interior angles are less than 180°, and none of its sides “push inward”.
If you extend any side, the line does not pass through the inside of the polygon.

Concave Polygon

A concave polygon is a polygon that has at least one interior angle greater than 180°.
It looks like it has a “dent” or an inward bend.

Regular Polygon

A regular polygon is a polygon where all sides have the same length and all interior angles have the same measure.

Interior Angles of a Polygon

The interior angles of a polygon are the angles inside the shape, formed by its sides.

Sum of Interior Angles

For triangles, the sum of the interior angles is always 180°.

For any polygon with n sides, the sum of its interior angles is:

(n - 2) · 180º

This works because any polygon can be divided into (n - 2) triangles.

Activities

How many diagonals has an heptagon? And a 100-agon?
Find out the sum of all the angles of an hexagon? What about a 100-agon?
Page 82, exercise 2.
Page 83, exercise 4.

Similarity

Which one is the olimpics logo?

Two shapes are similar (semejantes) if they have the same form but different size. You can say that if two shapes are similar, one of them can become the other by moving, flipping, rotating or zooming it.

If two shapes are similar:

Corresponding angles are the same.
Corresponding sides are proportional.

The ratio between sides of similar shapes is called ratio of similarity (razón de semejanza).

Group the following shapes by similarity

Look at the following cards. There are two of them that are similar. What is the ratio of similarity between the biggest and the smallest one? According to that ratio, find out the height of the small card.
The sides of a rectangle measure 6 cm and 8 cm. Is it similar to another rectangle measuring 15 cm by 24 cm? And to another one measuring 12 cm by 16 cm?
Tell if the following shapes are similar:

Let's practise similarity with the following activity.

Thales Theorem

If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.

Teorema de Thales en Geogebra.

Indirect measurements with similarity

How do you solve a problem using indirect measurement with shadows?

Exercises

1. The flagpole. A flagpole casts a shadow of 6 m at the same time a student of 1.6 m casts a shadow of 2.4 m. Calculate the height of the flagpole.

2. The school building. The shadow of the school building is 18 m long. At the same moment, a 1.5 m tall teacher casts a 3 m shadow. How tall is the school building?

3. The tree in the playground. A tree in the playground is impossible to measure directly. Its shadow measures 10 m. Nearby, a 1.4 m tall student casts a 2 m shadow. Find the height of the tree.

4. Measuring a tower with a mirror. A small mirror is placed on the ground 1.2 m from your feet. You can see the top of a tower in the mirror ,that is 18 m away from its foot. Your height is 1.6 m. Calculate the height of the tower.

5. Streetlight. A streetlight casts a shadow of 12 m. At the same time, a 1.7 m tall person casts a shadow of 2.5 m. How tall is the streetlight?

6. Cliff from a distance. From a point on the ground, a 0.5 m stick casts a shadow of 0.8 m. At the same time, a cliff casts a shadow of 40 m. Calculate the height of the cliff.

7. The inaccessible wall. A wall is too high to be measured directly. A 2 m tall ladder leaning vertically casts a 1.2 m shadow, while the wall casts a 6 m shadow. Find the height of the wall.

8. Using parallel lines (Thales’ theorem). Two parallel lines are cut by two transversals. On the first transversal, the segments measure 3 m and 5 m. On the second transversal, one corresponding segment measures 4 m. Find the length of the other corresponding segment.

9. The mountain cable. From a viewpoint, a 1.5 m vertical pole appears aligned with the top of a mountain when viewed from a certain point. The pole is 5 m away, and the mountain base is 500 m away. Estimate the height of the mountain.

10. Shadow at sunset (challenge). At sunset, a tower casts a shadow of 45 m. A student measuring 1.65 m casts a shadow of 2.2 m.

a) Calculate the height of the tower.

b) Explain why the triangles used are similar.

La escala de un mapa/The scale of a map

The scale of a map is simply the ratio of similarity between two similar “figures”: the real world and the map. When we say that a map has a scale of 1 : 1 000 000, we mean that reality is 1000000 times larger than the map.

Therefore, to convert measurements from that map into real-life measurements, we must multiply them by 1000000.

The relationship between measurements on a map (M), real-life measurements (R), and the scale factor (e) is given by the following formula:

M · e = R

It is important to note that scale is a dimensionless quantity. It converts centimetres to centimetres or metres to metres, but it has no units of its own.

Ejercicios

Draw a map of your bedroom. Escale 1:20.
3. Mira el mapa que hay en tu clase y calcula la distancia en línea recta de Granada a París.
4. Haz un vídeo donde midas la altura de un objeto muy alto. Podrás ganar hasta 3+. Puedes hacerlo en grupo, si lo deseas, de un máximo de 3 componentes.
6. If your notebook was a tablet. How many inches would it be? Remember, 1” = 2,54 cm
7. ¿Cuál es el parque más grande de Granada? Hay quien dice que es el Parque Federico García Lorca y otros, que es el Parque Tico Medina. Búscalos en un mapa y realiza las siguientes tareas:
a) Haz un dibujo a escala en tu cuaderno de los dos. Indica cuál es la escala del mismo.
b) Calcula el perímetro de ambos parques atendiendo a la escala y tu propio dibujo.
c) Calcula el área de los dos parques.
d) Da una respuesta a la pregunta inicial, de acuerdo con tus cálculos.
Avanzado: ¿Son semejantes los rectángulos exterior e interior de un marco?

Relación entre semejanza de figuras y superficie

Observa el campo de Los Cármenes en Google Maps. Calcula su superficie en el mapa y en la realidad. Compara las superficies. ¿Qué pasa?.

Efectivamente, el factor de escala no es el mismo, sino que se ha elevado al cuadrado.

Pensemos ahora en un cubo de Rubik 2x2x2 y otro 3x3x3 hecho con los mismos cubitos. ¿Qué relación existe entre las superficies de sus caras? ¿y entre sus volúmenes? ¿y entre sus pesos? Ten en cuenta que el peso (o la masa, para hablar con propiedad) depende directamente del volumen de los objetos, según la siguiente relación: m = V · d, donde d es la densidad del objeto en cuestión.

Calcula el peso de la figura grande de Baymax teniendo en cuenta los datos del dibujo siguiente.

2. Se quiere dibujar en la pared del aula un "Bart Simpson" gigante, que llegue hasta el techo. Se usará como modelo, el siguiente dibujo:

Fractales

Definición.

Peine de Cantor.
Dibuja un árbol fractal.
Dibuja el triángulo de Sierpinski.
Dibuja el copo de nieve de Koch.
Dibuja el árbol de Pitágoras.
Mostrar un fractal en 3D: hidrangea y el que se hace con tijeras.

Soluciones

Enlaces

Criterios de evaluación y saberes básicos

MAT.1.1.3.Obtener las soluciones matemáticas en problemas de contextos cercanos de la vida cotidiana, activando los conocimientos necesarios, aceptando el error como parte del proceso.

MAT.1.2.1. Comprobar, de forma razonada la corrección de las soluciones de un problema, usando herramientas digitales como calculadoras, hojas de cálculo o programas específicos.

MAT.1.6.1.Reconocer situaciones en el entorno más cercano susceptibles de ser formuladas y resueltas mediante herramientas y estrategias matemáticas, estableciendo conexiones entre el mundo real y las matemáticas y usando los procesos inherentes a la investigación científica y matemática: inferir, medir, comunicar, clasificar y predecir, aplicando procedimientos sencillos en la resolución de problemas.

MAT.1.6.2.Analizar conexiones coherentes entre ideas y conceptos matemáticos con otras materias y con la vida real y aplicarlas mediante el uso de procedimientos sencillos en la resolución de problemas en situaciones del entorno cercano.

MAT.1.7.2.Esbozar representaciones matemáticas utilizando herramientas de interpretación y modelización como expresiones simbólicas o gráficas que ayuden en la búsqueda de estrategias de resolución de una situación problematizada.

Preguntas del examen

Questions about polynomials. Name, angles, vertices, sum of its angles, number of diagonals, perimeter...
Identify similar shapes. Calculate the ratio.
Measuring heights indirectly. Shadow or mirror methods.
Maps. Questions about a map: calculating the scale, the real-life measurements, or the measurements on the map knowing the other two elements.
Hand in a plan of your room. Include the real dimensions of the room and the scale. You can see the appropriate symbols for its representation in the infographic above.

Google Sites

Report abuse