Using the Pythagorean Theorem in a Word Problem

Word problems often hide a right triangle inside a practical situation: a ladder against a wall, a diagonal across a rectangular field, or a ramp rising over a horizontal distance. The key first step is recognizing that the theorem applies only when the triangle has a right angle.

Identify the hypotenuse before calculating

The hypotenuse is the side opposite the 90° angle and is always the longest side. The other two sides are the legs. If the unknown is the hypotenuse, add the squared legs and take the square root. If a leg is unknown, subtract the square of the known leg from the square of the hypotenuse, then take the square root.

Example: ladder against a wall

A 13-foot ladder rests against a wall, with its base 5 feet from the wall. The ladder is the hypotenuse, so let h be the vertical height. Then h² + 5² = 13²; h² = 169 − 25 = 144; h = 12 feet. The ladder reaches 12 feet up the wall.

Example: diagonal distance

A rectangular garden is 9 meters wide and 12 meters long. The diagonal is the hypotenuse: d² = 9² + 12² = 81 + 144 = 225, so d = 15 meters. This is a common 3-4-5 ratio scaled by 3, a useful reasonableness check.

A reliable setup process

• Sketch and label the right angle and the three side lengths.
• Mark the side opposite the right angle as c, the hypotenuse.
• Convert all measurements to the same unit.
• Write the theorem with the unknown in the correct position.
• Solve, keep the positive length, and round only as requested.
• State what the number measures and include units.

Common mistakes

• Using the theorem on a triangle that is not right-angled.
• Calling the side along the ground the hypotenuse when it is a leg.
• Adding squares when the unknown is a leg instead of subtracting.
• Forgetting the square root after solving for x².
• Reporting a negative length or a result longer than the hypotenuse when solving for a leg.

Identify the right triangle and its hypotenuse

The Pythagorean theorem applies only to a right triangle: a² + b² = c², where a and b are the legs that meet at the right angle and c is the hypotenuse opposite it. The hypotenuse is always the longest side. In a word problem, first sketch the situation and mark the 90° corner; do not simply choose the side described as “across” or “diagonal” without checking its position.

A 5-foot ladder reaches a wall, with its base 3 feet from the wall. The ladder is the hypotenuse, so the height is h² + 3² = 5². Thus h² = 25 − 9 = 16 and h = 4 feet. The negative square root is rejected because a length cannot be negative. A 3-4-5 triangle is a useful check, but the equation still explains why those lengths fit.

Choose addition or subtraction from the unknown

When the unknown is the hypotenuse, add the squares of the legs and take the square root. If the unknown is a leg and the hypotenuse is known, subtract the known leg’s square from the hypotenuse’s square. A common setup error is adding when solving for a leg. For a triangle with hypotenuse 13 and one leg 5, the other leg is √(13² − 5²) = √144 = 12, not √194.

Use the theorem in real settings

The theorem models the shortest straight-line distance across a rectangular field. If a person walks 6 meters east and then 8 meters north, the direct distance from start to finish is √(6² + 8²) = 10 meters. The path walked is 14 meters; the displacement is 10. A diagram helps distinguish total route length from the diagonal being asked for.

For a rectangular screen, room, or plot, the diagonal and two perpendicular sides form a right triangle. For a map problem, the east-west and north-south distances may be the legs if the directions are perpendicular and use consistent units. Convert units before squaring; adding square feet to square inches produces a meaningless result.

Check whether a triangle is possible

The hypotenuse must exceed either leg. If a calculation gives a negative value under the square root or a hypotenuse shorter than a leg, recheck which side is opposite the right angle and whether the stated measurements are compatible. Exact square roots such as √50 can be simplified to 5√2; use a decimal approximation only when requested or appropriate. Round at the end so intermediate rounding does not reduce accuracy.

• Mark the right angle; c is the opposite and longest side.
• Write a² + b² = c² before substituting values.
• Add squares for a missing hypotenuse; subtract for a missing leg.
• Keep all lengths in one unit and attach that unit to the final answer.
• Check that the result is positive and geometrically plausible.

Connect the theorem to a diagram

Draw the right angle and label the two perpendicular legs before writing a² + b² = c². If a ramp rises 1.2 meters over a horizontal run of 5 meters, its straight length is √(1.2² + 5²) = √26.44, about 5.14 meters. The vertical rise is not the ramp length; the sketch identifies which side is the hypotenuse.

The converse of the Pythagorean theorem can test whether given lengths form a right triangle. If the longest side is c and a² + b² = c², the angle opposite c is a right angle. For 6, 8, and 10, 36 + 64 = 100, so the triangle is right. Always test the longest side as the possible hypotenuse.

Exam takeaway

Find the right angle first. The opposite side is c; use a² + b² = c², then interpret the positive solution with the correct units.

Use the concept in a question

Use a² + b² = c² only for a right triangle, with c opposite the right angle as the hypotenuse. If a 5-foot ladder reaches 12 feet up a wall, the ground distance is not found by treating 12 as the hypotenuse: the ladder is the hypotenuse, so 5² must be the shorter leg and the setup is impossible because 5 is shorter than 12. A consistent example is a 13-foot ladder reaching 12 feet high: ground distance = √(13² − 12²) = 5 feet. Sketch and label the right angle before choosing the formula, then keep units consistent and check that the hypotenuse is the longest side.