Applying the theorem to word and figure problems

Word and figure problems rarely announce "use the Baudhayana–Pythagoras theorem" outright. Instead they describe a ladder, a diagonal, a pole, or a corner, and it is up to you to notice the right triangle sitting inside the description. There are four recurring setups worth memorising:

• Ladder against a wall. The wall and the ground meet at a right angle; the ladder itself is the hypotenuse, the height it reaches is one leg, and the distance of its foot from the wall is the other leg.
• Diagonal across a rectangle. The two sides of the rectangle (a field, a screen, a room) are the two legs; the diagonal joining opposite corners is the hypotenuse.
• Checking a right angle. Given three measured lengths, test whether $a^2+b^2=c^2$ holds with $c$ as the largest length. If it holds, the angle opposite $c$ is exactly $90°$ — this is the converse of the theorem, and it is how carpenters and masons check square corners without a protractor.
• Shortest straight-line distance. When two points differ by a known horizontal amount and a known vertical amount (on a map, a grid, or a coordinate diagram), the straight-line distance between them is the hypotenuse of the right triangle formed by those two differences.

Once the right triangle is identified, the algebra is always the same relation, just solved for a different letter:

where: $a$ and $b$ are the two legs of the right triangle (the sides meeting at the right angle), and $c$ is the hypotenuse (the side opposite the right angle, always the longest side). Depending on which quantity is unknown, the same relation is rearranged as

where: the square root undoes the squaring, so whichever side is missing is found by isolating its square and taking the positive square root (a length is never negative). Before writing any equation, it helps to ask the figure three quick questions: Where exactly is the right angle? Which side is opposite it — that's the hypotenuse? What am I actually being asked to find? These three questions turn almost any word problem into a direct plug-in. It is also worth noticing that once one unknown side is found, a follow-up question — a total distance walked, an amount of rope saved, a perimeter — is usually just simple addition or subtraction on values you already have, as the worked example below shows.

Problem: A rectangular park measures $30\text{ m}$ by $40\text{ m}$. Instead of walking along two sides, a student walks straight across the diagonal. How much shorter is this diagonal path compared to walking along the two sides?

Step 1. Sketch the triangle: the two sides of the rectangle, $30\text{ m}$ and $40\text{ m}$, are the legs of a right triangle formed at one corner; the diagonal joining the opposite corners is the hypotenuse.

Step 2. Apply the theorem with $a=30$, $b=40$: $a^2+b^2=c^2$ gives $30^2+40^2=c^2$, that is $900+1600=c^2$, so $c^2=2500$.

Step 3. Take the square root: $c=\sqrt{2500}=50\text{ m}$, the length of the diagonal path.

Step 4. Find the distance along the two sides: $30+40=70\text{ m}$.

Step 5. Find the difference between the two routes: $70-50=20\text{ m}$.

Walking the diagonal saves the student $20\text{ m}$ compared to walking along the two sides. Note $30,40,50$ is a multiple of the primitive triple $3,4,5$, a useful pattern to recognise instantly.

Competitive stretch

A gardener wants to check whether a corner of a rectangular flower bed is truly a right angle. She measures $9\text{ m}$ and $12\text{ m}$ along the two sides from the corner and finds the straight-line distance between those two marked points is $15\text{ m}$. Since $9^2+12^2=81+144=225=15^2$, the relation $a^2+b^2=c^2$ holds, so by the converse of the theorem the angle at that corner is exactly $90°$ — the corner is square.

• Not drawing a sketch first. Word problems become far easier once the right triangle is drawn and the hypotenuse labelled; skipping this step often means plugging numbers into the wrong position in the formula.
• Treating the ladder or diagonal as a leg instead of the hypotenuse. A ladder, a diagonal, or any "straight-line distance across" is almost always the longest side and therefore the hypotenuse, not one of the legs.
• Adding the two known sides directly instead of using the theorem. For a right-triangle situation you cannot just add lengths — $30+40$ is not the diagonal — you must square, add, and take the square root.
• Forgetting units or mixing them. Convert all lengths to the same unit (cm to m, or vice versa) before substituting into the theorem, and state the final answer with the correct unit and in the context of the question asked.

Ask the tutor: Stuck sketching the right triangle hidden in a ladder, diagonal, or right-angle-checking problem — ask the AI tutor to walk through labelling the hypotenuse and legs with you.