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The Pythagorean theorem
Dynamic tasks on legs, hypotenuse, rectangle diagonals, coordinate distance and practical applications.
Questions: 10Estimated time: 20 minutes
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Question 1out of 10
Find the distance between (1, -4) and (10, 8).
Horizontal difference dx = x2 − x1 and vertical difference dy = y2 − y1. Distance = √(dx² + dy²). Differences may be negative, but their squares are nonnegative.
Question 2out of 10
A destination can be reached by walking 7 m east and 24 m north, or directly. Which route is shorter?
The route with a turn is a + b; the direct route is √(a² + b²). When both lengths are positive, the direct route is shorter. If a difference is needed, subtract the direct length from a + b.
Question 3out of 10
Which set of three side lengths can form a right triangle?
Treat the longest side as c and check a² + b² = c². If this holds, the converse Pythagorean theorem shows the triangle is right-angled.
Question 4out of 10
A rectangle has sides 6 cm and 8 cm. Find its diagonal.
Perpendicular lengths form the legs of a right triangle. By Pythagoras, hypotenuse c = √(a² + b²). Square the lengths, add and take the square root; do not simply add the lengths.
Question 5out of 10
The legs of a right triangle are 12 cm and 16 cm. Find the hypotenuse.
Perpendicular lengths form the legs of a right triangle. By Pythagoras, hypotenuse c = √(a² + b²). Square the lengths, add and take the square root; do not simply add the lengths.
Question 6out of 10
Which set of three side lengths can form a right triangle?
Treat the longest side as c and check a² + b² = c². If this holds, the converse Pythagorean theorem shows the triangle is right-angled.
Question 7out of 10
A destination can be reached by walking 7 m east and 24 m north, or directly. Which route is shorter?
The route with a turn is a + b; the direct route is √(a² + b²). When both lengths are positive, the direct route is shorter. If a difference is needed, subtract the direct length from a + b.
Question 8out of 10
If a triangle satisfies a² + b² = c², what follows from the converse Pythagorean theorem?
Treat the longest side as c and check a² + b² = c². If this holds, the converse Pythagorean theorem shows the triangle is right-angled.
Question 9out of 10
The foot of a ladder is 32 m from a wall and its top reaches 60 m high. How long is the ladder?
Perpendicular lengths form the legs of a right triangle. By Pythagoras, hypotenuse c = √(a² + b²). Square the lengths, add and take the square root; do not simply add the lengths.
Question 10out of 10
A right triangle has hypotenuse 50 cm and one leg 48 cm. Find the other leg.
From a² + b² = c², the missing leg is a = √(c² − b²). Subtract the known leg squared from the hypotenuse squared and take the square root.
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Sample questions
- The legs of a right triangle are ___ cm and ___ cm. Find the hypotenuse.
- A right triangle has hypotenuse ___ cm and one leg ___ cm. Find the other leg.
- Which set of three side lengths can form a right triangle?
- A rectangle has sides ___ cm and ___ cm. Find its diagonal.
- Find the distance between (___, ___) and (___, ___).
The Pythagorean theorem
In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse: a² + b² = c². The hypotenuse lies opposite the right angle and is always the longest side. Find a missing leg by subtracting the known leg squared from c².
Applications
The theorem applies to rectangle diagonals, coordinate distances and practical length problems. Confirm that the triangle is right-angled, identify the hypotenuse and only then substitute values. The answer uses units of length, not square units.
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