The Pythagorean Theorem

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.

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

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

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

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 $2.4$ meters and the distance from the base of the post to the anchor is $1.8$ meters. 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?

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.

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

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

• a) The actual height of the garden.

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

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.

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 $4\text{ meters}$ and its maximum height is $3\text{ meters}$. At a distance of $1.6\text{ meters}$ 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.

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

• 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 triangles are formed. Calculate the length of these imaginary lines (hypotenuses) for both buildings.

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.

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.

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 fiber 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).

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.

A courier service uses cardboard boxes in the shape of a 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 the Pythagorean 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.

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 $\text{cm}^2$?

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

1. Calculate the slant height (apothem of the pyramid) using the Pythagorean 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.