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Mathematics · 05 / 23

Right triangles: choose the ratio

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

Need a foundation?Pythagoras: lengths and right-angle checksVectors: displacement, not location
UnderstandSee an exampleGuided exercise3 progressive challenges

The principle, simply

The hypotenuse is opposite the right angle. Pythagoras connects lengths; sine, cosine and tangent connect an acute angle to side ratios. Relative to that angle: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent. Opposite and adjacent depend on the chosen angle.

c=sqrt(a^2+b^2)
THE PICTURE TO REMEMBERRight triangles: choose the ratio
  1. 01Reference angle
  2. 02Name the sides
  3. 03Choose the ratio
  4. 04Check angle units

WORKED EXAMPLE

From principle to reasoning

Perpendicular sides 3 and 4 give hypotenuse √(9 + 16) = 5. For the angle opposite side 3, sin α = 3/5.

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

An angle has opposite side 6 and hypotenuse 10. Find its sine.

One hint

Name the sides relative to the angle.

Reveal and understand the solution

sin α = 6/10 = 0.6. To find the angle, arcsin in degrees gives about 36.87°.

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

A right triangle has a 30° angle and 10 cm hypotenuse. Find the opposite side.

A hint to get started

Use sine rather than cosine.

Step-by-step solution
  1. sin(30°) = opposite/10.
  2. sin(30°) = 0.5, so the opposite side is 5 cm.
  3. This is shorter than the hypotenuse.

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

03 · Apply the methodLink to this exercise

Opposite and adjacent sides are 3 cm and 4 cm. Write the angle calculation.

A hint to get started

Opposite/adjacent is tangent.

Step-by-step solution
  1. tan(θ) = 3/4.
  2. θ = arctan(3/4) ≈ 36.87°.
  3. Use degree mode when a degree answer is requested.

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

04 · Go furtherLink to this exercise

A 5 m ladder reaches 4 m up a wall. How far is its foot from the wall?

A hint to get started

Wall, ground and ladder form a right triangle.

Step-by-step solution
  1. The 5 m side is the hypotenuse.
  2. Distance² = 5² − 4² = 9.
  3. Distance = 3 m, taking the positive length.

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

The trap to avoid

Pythagoras requires a right angle; use a consistent degree or radian mode.

Without notes, can you…

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

Choosing a state saves it in this browser. This is your assessment, not a grade.

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