The Pythagorean Theorem

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

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

A lighthouse projects a light that reaches a boat. The lighthouse has a height of $30$ feet and the boat is at a horizontal distance of $40$ feet from the base of the lighthouse. What is the direct distance between the lighthouse bulb and the boat?

Find the length of the unknown side ($c$) in the following right-angled triangle, knowing that the hypotenuse measures $13$ in and one of the shorter sides measures $5$ in.

Check if a triangle whose sides measure $7$ in, $10$ in and $12$ in is a right-angled triangle. To do this, verify if the equality of Pythagoras' Theorem is satisfied between its sides.

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 that measures exactly $82\text{ cm}$.

Use the Pythagorean Theorem to determine if the tube fits inside the box (placing it diagonally joining two opposite vertices of the interior space) and justify your answer with the necessary calculations.

An open stepladder has a vertical height of $2,4$ feet. If the distance between the legs resting on the ground is $1,4$ feet, how long is each of the arms of the ladder? Hint: Notice that the height divides the ladder into two equal right-angled triangles.

Calculate the perimeter of a rhombus whose diagonals measure $16$ in and $12$ in. Remember that the diagonals of a rhombus are perpendicular and intersect at their midpoint, forming four right-angled triangles.

Calculate the area of a regular hexagon with a side 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.

To secure a volleyball net in the school playground, a tension cable has been placed from the top of a post to an anchor in the ground. The post has a height of $2,4$ feet and the distance from the base of the post to the anchor is $1,8$ feet. If we want to put an identical cable on another post that is a scale enlargement of $k = 1,5$ of the first one, how long will the new tension cable 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 shortest 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 small shelf has two shorter sides of $30\text{ cm}$ and $40\text{ cm}$. The large shelf has a perimeter of $240\text{ cm}$. How long is the longest side (hypotenuse) of the large shelf?

In a park there is a skate ramp that, seen in profile, forms a right-angled triangle. The base of the ramp measures $4\text{ meters}$ and the maximum height is $3\text{ meters}$. At a distance of $1,6\text{ meters}$ from the base, a vertical reinforcement support has been placed.

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

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

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

• a) What is the height of building B?

• b) If we join the highest part of each building with the end of its shadow, two right-angled triangles are formed. Calculate the length of those imaginary lines (hypotenuses) for both buildings.

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

1. What minimum length must the cable have?

2. If a technician is positioned 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 similar triangles).

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

• a) The real height of the garden.

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

An advertising company has created a rhombus-shaped canvas to cover the facade of a building. The diagonals of the rhombus measure $16\text{ m}$ and $12\text{ m}$. They want to make a scale replica for an office whose scale factor with respect to the original is $k = 0,25$.

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

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

1. Calculate the height of a lateral face (slant height of the pyramid) using Pythagoras.

2. If we build a smaller similar pyramid with a scale factor of $k = 0,5$, calculate the volume of the small pyramid knowing that the volume of the original is $V = \frac{1}{3} \cdot \text{Base area} \cdot \text{Height}$.

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

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

• b) Once the height is found, use Pythagoras' Theorem to calculate how long the two shorter sides of the original triangle are.

An architect has designed a glass window in the shape of an isosceles triangle whose base measures $12\text{ m}$ and its 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 glass window using the Pythagorean Theorem and, subsequently, use similar triangles to find the length of the horizontal crossbar.

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

1. Calculate the radius of this circle (apothem of the hexagon) using the Pythagorean Theorem.

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

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

Calculate 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.