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Mathematics · 08 / 23

Pythagoras: lengths and right-angle checks

Middle school · Worked example and 4 exercises with solutions, at your pace.

UnderstandSee an exampleGuided exercise3 progressive challenges

The principle, simply

In a right triangle, the hypotenuse squared equals the sum of squares of the perpendicular sides. It lies opposite the right angle and is longest. The converse checks for a right angle when the largest side squared equals the other squares combined. Lengths are positive and use one unit.

c=sqrt(a^2+b^2)
THE PICTURE TO REMEMBERPythagoras: lengths and right-angle checks
  1. 01Right angle
  2. 02Hypotenuse
  3. 03Squares and root
  4. 04Length and unit

WORKED EXAMPLE

From principle to reasoning

Sides 5 and 12 give c²=169 and c=13. With hypotenuse 10 and side 6, the other side is √(100−36)=8.

Solve a problem in this subject

  1. Define the objects

    State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.

  2. Choose a transformation

    Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.

  3. Show the reasoning

    Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.

  4. Test and conclude

    Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.

YOUR TURN

01 · Try before checking

Are sides 7, 24 and 25 a right triangle?

One hint

Compare 25² with 7²+24².

Reveal and understand the solution

Both equal 625. The converse proves a right triangle with hypotenuse 25.

Compare method, conditions and answer; identify the exact part to revisit.

IMPROVE THROUGH PRACTICE

Three additional challenges with solutions

Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.

Reflect after practising
02 · Recognise and understandLink to this exercise

Perpendicular sides measure 6 and 8 cm. Find the hypotenuse.

A hint to get started

Add the squares of the shorter sides.

Step-by-step solution
  1. c² = 6² + 8² = 100.
  2. Positive length c is 10 cm.
  3. Do not use 6 + 8: the theorem uses squares.

Which step did you miss? Retry without help, then explain why the method works.

03 · Apply the methodLink to this exercise

Is a triangle with sides 5, 12 and 13 cm right-angled?

A hint to get started

Test the square of the longest side.

Step-by-step solution
  1. 5² + 12² = 25 + 144 = 169.
  2. 13² = 169, so equality holds.
  3. The converse proves a right angle opposite the 13 cm side.

Which step did you miss? Retry without help, then explain why the method works.

04 · Go furtherLink to this exercise

A rectangle is 9 m by 12 m. Find a cable joining opposite corners.

A hint to get started

The diagonal is the hypotenuse of a right triangle.

Step-by-step solution
  1. d² = 9² + 12² = 225.
  2. d = 15 m.
  3. This is the direct distance; following the edges would need 21 m.

Which step did you miss? Retry without help, then explain why the method works.

The trap to avoid

Use squares, not a sum of lengths. First establish a right angle or test the converse.

Without notes, can you…

  • Explain the principle in your own words?
  • Rebuild the diagram and its connections?
  • Solve the exercise and justify the method?

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