Exercises: Pythagoras' Theorem

In a football training session, a player runs in a straight line from one corner of the pitch to the opposite corner. If the pitch is $80$ metres long and $60$ metres wide, what distance does the player cover in their diagonal run?

An open stepladder has a vertical height of $2.4$ metres. If the distance between the legs resting on the ground is $1.4$ metres, how long is each leg of the stepladder? Hint: Notice that the height divides the ladder into two congruent right-angled triangles.

Check whether a triangle with sides measuring $7$ cm, $10$ cm and $12$ cm is a right-angled triangle. To do this, check if Pythagoras' theorem holds true for its sides.

We want to attach a cable from the top of a $12$ metre high mobile phone mast to an anchor point on the ground, $9$ metres from the base of the mast. How long must the cable be so that it is pulled completely taut?

Find the unknown side length ($c$) in the following right-angled triangle, knowing that the hypotenuse is $13$ cm long and one of the legs is $5$ cm long.

Television screens are measured by their diagonal in inches. If a television has a width of $32$ cm and a height of $24$ cm, how many centimetres is its diagonal?

Calculate the perimeter of a rhombus whose diagonals measure $16$ cm and $12$ cm. Remember that the diagonals of a rhombus are perpendicular and bisect each other, forming four right-angled triangles.

A lighthouse projects a beam of light towards a boat. The lighthouse is $30$ metres high and the boat is at a horizontal distance of $40$ metres from the base of the lighthouse. What is the direct distance between the light bulb of the lighthouse and the boat?

To calculate the length of the diagonal of a rectangular TV screen that measures $40\text{ cm}$ wide and $30\text{ cm}$ high, Aoife writes the following procedure in her notebook:

$d = \sqrt{40^2 + 30^2} = \sqrt{40^2} + \sqrt{30^2} = 40 + 30 = 70\text{ cm}$.

1. a) Explain in detail what algebraic error Aoife made with the square root and why the square root of a sum cannot be split into the sum of the square roots.
2. b) Without doing the correct calculation yet, reason geometrically why a result of $70\text{ cm}$ is impossible for that diagonal (remember the triangle inequality: the length of any side of a triangle must be less than the sum of the other two).
3. c) Correct Aoife's procedure and find the actual measurement of the screen's diagonal.

A school basketball court has official dimensions of $28\text{ m}$ long by $15\text{ m}$ wide. The Physical Education teacher asks their students to run in a straight line from one corner to the diagonally opposite corner.

1. a) Before doing any operations with square roots, make a reasoned estimate: justify why the diagonal distance covered must necessarily be greater than $28\text{ m}$ and less than $43\text{ m}$. Between which two whole numbers do you think the result will be?
2. b) Apply the Pythagorean Theorem to calculate the exact length of the diagonal, rounding the result to the nearest hundredth of a metre.
3. c) Compare the value obtained with your previous estimate and critically assess whether the result obtained makes practical sense on a sports court.

To secure a volleyball net in the school playground, a guy wire has been attached from the top of a post to an anchor on the ground. The post has a height of $2.4$ metres and the distance from the base of the post to the anchor is $1.8$ metres. If we want to attach an identical wire to another post that is a scaled enlargement by a factor of $k = 1.5$ of the first one, how long will the new guy wire be?

A rectangular smartphone screen has a height of $12\text{ cm}$ and a width of $5\text{ cm}$. A tablet is similar to the smartphone and its shorter side measures $15\text{ cm}$. First, calculate the diagonal of the smartphone using Pythagoras' theorem, and then use the scale factor to find the diagonal of the tablet.

A carpenter is designing two similar triangular shelves for the corners of a room. The smaller shelf has two shorter sides (legs) of $30\text{ cm}$ and $40\text{ cm}$. The larger shelf has a perimeter of $240\text{ cm}$. How long is the longest side (hypotenuse) of the larger shelf?

In a park, there is a skate ramp whose side view forms a right-angled triangle. The base of the ramp measures $4\text{ metres}$ and its maximum height is $3\text{ metres}$. At a distance of $1.6\text{ metres}$ from the base, a vertical support beam has been placed for reinforcement.

• a) Calculate the total length of the ramp (the sloped part).

• b) Using similar triangles, calculate the height of the vertical support beam.

A garden is in the shape of an isosceles triangle. Its two equal sides measure $13\text{ metres}$ and the third side (the base) measures $10\text{ metres}$. We want to build a scale model of this garden that is a reduction where the base of the model is $20\text{ cm}$. Work out:

• a) The actual height of the garden.

• b) The area that the model will occupy in $\text{cm}^2$.

Two buildings cast shadows at the same time of day. Building A is $15\text{ metres}$ high and its shadow measures $12\text{ metres}$. Building B stands next to it and casts a shadow of $20\text{ metres}$.

• a) What is the height of Building B?

• b) If we join the top of each building to the tip of its shadow, two right-angled triangles are formed. Calculate the length of these imaginary lines (hypotenuses) for both buildings.

Calculate the area of a regular hexagon with a side length of $8\text{ cm}$. Remember that a regular hexagon can be divided into six equilateral triangles. You will need to use Pythagoras' theorem to find the apothem of the hexagon before calculating its total area.

In a right-angled triangle, the altitude to the hypotenuse divides it into two segments measuring $9\text{ cm}$ and $16\text{ cm}$.

• a) Calculate the altitude of the triangle using the Altitude Theorem ($h^2 = m \times n$, where $m$ and $n$ are the two segments of the hypotenuse).

• b) Once the altitude is found, use Pythagoras' theorem to calculate the lengths of the two legs of the original triangle.

A carpentry workshop needs to design a connecting piece that is shaped like a right triangle and whose hypotenuse measures exactly $\sqrt{65}\text{ dm}$.

1. a) Find two different pairs of positive integers $(a, b)$ that can be the lengths of the legs in decimetres. Prove algebraically, using the Pythagorean Theorem, that both pairs yield exactly the required hypotenuse.
2. b) If the workshop decided that the piece must be, in addition to a right triangle, an isosceles triangle (with both legs equal) keeping the hypotenuse of $\sqrt{65}\text{ dm}$, set up an equation and deduce what each of the legs should measure exactly. Can they measure a whole number of decimetres in this second case? Justify your answer.

During fire surveillance tasks in the Sierra de Guadarrama, a forestry observation post of the INFOMA plan is located at an altitude of $1900\text{ m}$. The lookout detects a column of smoke whose position on the cartographic map is located $1200\text{ m}$ to the east and $900\text{ m}$ to the north of the tower, but in a valley located at a lower elevation, at an altitude of $1100\text{ m}$.

1. a) Model the situation on the horizontal plane and calculate the straight-line distance on the map (horizontal distance) separating the observation post from the column of smoke.
2. b) Model the situation in three dimensions, taking into account the altitude difference between the tower and the fire. Calculate the actual direct distance that a reconnaissance drone would cover in a spatial straight line from the observation post to the incipient fire.

A courier service uses cardboard boxes in the shape of a cuboid (rectangular prism). One of its largest boxes has dimensions of $60\text{ cm}$ long, $45\text{ cm}$ wide and $30\text{ cm}$ high. A customer needs to send a rigid tube measuring exactly $82\text{ cm}$.

Use Pythagoras' theorem to determine whether the tube fits inside the box (placing it diagonally connecting two opposite vertices of the internal space) and justify your answer with the necessary calculations.

An architect has designed a window in the shape of an isosceles triangle whose base measures $12\text{ m}$ and whose equal sides measure $10\text{ m}$ each. For lighting reasons, they need to place a horizontal crossbar exactly halfway up the height of the triangle.

First calculate the total height of the window using Pythagoras' theorem, and then use similar triangles to find the length of the horizontal crossbar.

An advertising company has created a rhombus-shaped tarpaulin to cover the facade of a building. The diagonals of the rhombus measure $16\text{ m}$ and $12\text{ m}$. They want to manufacture a scale replica for an office whose scale factor with respect to the original is $k = 0.25$.

Calculate the perimeter of the original tarpaulin (using Pythagoras' theorem to find the side length) and, from that data, determine the perimeter and area of the small replica without needing to calculate its sides one by one.

A drone is flying over flat ground at a height of $120\text{ m}$. At a given moment, its camera detects the top of a control tower. The sensors indicate that the straight-line distance (diagonal) from the drone to the top of the tower is $150\text{ m}$. If we know that the shadow of the tower at that same instant measures $18\text{ m}$, calculate:

1. The horizontal ground distance from the drone to the line directly above the tower.

2. The height of the control tower using similarity between the tower and its shadow, knowing that the sun's rays strike at an angle such that a pole of $2\text{ m}$ casts a shadow of $1.2\text{ m}$.

A garden is in the shape of a regular hexagon with side length $8\text{ m}$. A circular irrigation system is to be installed that covers exactly up to the midpoints of each side (that is, the circle inscribed in the hexagon).

1. Calculate the radius of this circle (apothem of the hexagon) using Pythagoras' theorem.

2. If a plan of the garden is drawn to a scale of $1:200$, what will be the area of the hexagon on the plan in $\text{cm}^2$?

Two vertical buildings are separated by a distance of $16\text{ m}$. Building A is $24\text{ m}$ tall and building B is $36\text{ m}$ tall. A fibre optic cable is to be stretched from the roof of building A to the roof of building B.

1. What is the minimum length the cable must have?

2. If a technician stands on the cable at a horizontal distance of $4\text{ m}$ from building A, at what height above the ground are they at that point? (Use triangle similarity).

There is a regular square pyramid whose base is a square with side length $10\text{ cm}$. The height of the pyramid (from the centre of the base to the top vertex) is $12\text{ cm}$.

1. Calculate the slant height (apothem of the pyramid) using Pythagoras' theorem.

2. If we build a similar smaller pyramid with a scale factor of $k = 0.5$, calculate the volume of the small pyramid knowing that the volume of the original is $V = \frac{1}{3} \times \text{Base area} \times \text{Height}$.

In a park, a swing is formed by two metal posts that cross each other forming an "X". The feet of the posts are $3\text{ m}$ apart on the ground. The posts measure $5\text{ m}$ each, are tilted symmetrically (they make the same angle with the ground) and cross at a height of exactly $2\text{ m}$ above the ground.

Work out the distance between the top ends of the posts using the properties of similarity (triangles in Thales' position) and Pythagoras' theorem to find the necessary auxiliary distances.

In a scale model of a playground, a straight scooter ramp is shaped like a right triangle with a base of $12\text{ cm}$ and a height of $5\text{ cm}$. The actual ramp built in the park is an enlarged similar figure with a scale factor of $k = 20$. To determine the length of the sloping surface of the actual ramp (the hypotenuse), two students propose different approaches:

• Seán's method: First, calculate the hypotenuse of the model using the Pythagorean Theorem and then multiply that measurement by the scale factor $k = 20$.

• Aoife's method: First, multiply the two dimensions of the model by the factor $k = 20$ to find the actual base and height in the park, and then apply the Pythagorean Theorem to those larger values.

1. a) Fully develop both methods step by step and check whether both lead to the same numerical result in metres.
2. b) If you had to solve this exercise on an exam without using a calculator, which of the two methods would you choose and why? Justify your choice by analysing the computational difficulty of the calculations involved and relying on the properties of similarity.

The Cork fire brigade has a rescue ladder with a fixed length of $6.5\text{ m}$. To reach a balcony, they rest the top end against the vertical façade of a building and the base on a horizontal pavement, at a distance of $2.5\text{ m}$ from the wall.

1. a) Model the situation with a right triangle and calculate what height on the façade the top end of the ladder reaches in that initial position.
2. b) To clear a cycle lane, the firefighters move the base of the ladder $0.9\text{ m}$ closer to the wall (leaving it at $1.6\text{ m}$ from it). Will the top end rise up the wall by exactly the same $0.9\text{ m}$ that the base has moved inward? Formulate a justified hypothesis before calculating, and then calculate the new height reached to test your hypothesis.
3. c) Compare the horizontal variation of the base with the vertical variation of the height and explain why, due to the quadratic relationship of the Pythagorean Theorem ($a^2 + b^2 = c^2$), the horizontal and vertical displacements do not keep a constant ratio and, in general, are not equal.