8. Measurements and Pythagorean Theorem

Una princesa de cuento quiere rescatar a un chico llamado Rapunzelete que se encuentra encerrado por un malvado brujo en una torre. La torre está rodeada de un peligroso foso lleno de cocodrilos y cantantes de reggaeton-trap. Ha comprobado que la ventana a la que desea acceder está a 17 metros de alto. Por otra parte, una amiga bombera le ha prestado para tan intrépida tarea una escalera de 15 metros. El foso, por su parte, tiene una anchura de 4.5 metros. ¿Tendrá el cuento un final feliz, esto es, podrá la princesa apoyar la escalera en la pared manteniendo sus extremidades (por los cocodrilos) y sus oídos (por los cantantes) intactos?

A Puzzle

Construye todos los cuadrados que puedas con las siguientes piezas:

Introductory situation

Have a look at the huge tablet in the video.

It is a 46-inch screen.

This article describes the features of that device.

Now imagine your desk is a tablet like this one. How many inches would it be?

Medidas directas y estimación de medidas / Direct measurements and estimation.
Errores. Precisión en la medida / Errors. Precision in measurements.
Teorema de Pitágoras / Pythagorean Theorem.
Utilización de los teoremas de Tales y Pitágoras para obtener medidas y comprobar relaciones entre figuras / Application of Pythagorean and Thales Theorems to measurements.

Measurements and estimations

If you use a tool to do it, that is a direct measurement. For example, you use a ruler to measure the side of a square.

Write three examples of direct measurements, with different magnitudes.

The alternative when you are not able or willing to make a direct measurement is to do indirect measurements or estimations.

You can practise estimations with the following "Hedge a bet" game.

Another example: estimating time.

Do the test ten times.

Which of them is the most accurate? Why is it?

The difference between the exact value and the measured value is the error.

E = |V - M|

The relative error is that number divided by the exact value. This relative error is usually given in a percent form.

As an example, if the exact value is 4.24 and the approximation is 4, then the absolute error is |4 - 4.24| = 0.24 and the relative error is 0.24/4.24 ~ 0.057 = 5.7 %.

Como los instrumentos de medida son imprecisos, siempre se comete algún error, aunque sea pequeño.

Exercise

Calculate the relative error of all the estimations we did in the game "Hedge a bet".

Practica con este juego la estimación de ángulos. Explica cómo funciona la puntuación.

Another puzzle

Solve the following puzzles (click here to get a printable version of them). What conclusion do you get?

The Pythagorean Theorem

In every right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

$a^2 + b^2 = c^2\!\,$

Exercises

Recomendaciones generales:

Haz un dibujo de la situación siempre que puedas.
Encuentra el triángulo rectángulo (busca el ángulo recto).
Averigua si te falta la hipotenusa o algún cateto y aplica el Teorema de Pitágoras.
Ofrece una respuesta apropiada a la pregunta y el contexto de la misma.

Find the unknown side for every triangle:
A ladder, 12 metres long, leans against a wall. The ladder reaches 10 metres up the wall. How far away from the foot of the wall is the foot of the ladder?
Santiago Bernabeu football pitch is 105 metres long and 70 metres wide. How long is the diagonal?
How long is the diagonal of the biggest square you can draw on your notebook? Calculate without measuring it directly. Use your ruler to calculate the error in your indirect measurement.
A plane flies from Granada due north for 120 km before turning due west and flying for a further 85 km and landing at a secret location. How far from Granada is the secret location?
At the moment, three towns, A, B and C, are joined by two roads, as in the diagram. The council want to make a road that runs directly from A to C. How much distance will the new road save? A mast on a sailboat is strengthened by a wire (called a stay), as shown on the diagram. The mast is 35 m tall and the stay is 37 m long. How far from the base of the mast does the stay reach?
A 4-metre ladder is put up against a wall. How far up the wall will it reach when the foot of the ladder is 1 m away from the wall? When it reaches 3.6 m up the wall, how far is the foot of the ladder away from the wall?
A pole, 8 m high, is supported by metal wires, each 8.6 m long, attached to the top of the pole. How far from the foot of the pole are the wires fixed to the ground?
How long is the line that joins the two coordinates A(10, 5) and B(1, 2)?
A rectangle is 4.5 cm long. The length of its diagonal is 5.8 cm. What is the area of the rectangle?
Two large trees, 5.5 m and 6.8 m tall, stand 12 m apart. A bird flies directly from the top of one tree to the top of the other. How far has the bird flown?
Have a look a this video. Which size is the wrap paper, at least, if the gift is a Rubik Cube?
Find the perimeter of a right isosceles triangle whose catetus are 20 m long.
Find x

13. Exercises from this PDF.