Right triangles and non-right triangles · Higher · Triángulos rectángulos y no rectángulos · Nivel avanzado
| English | Español |
|---|---|
| hypotenuse/haɪˈpɒtənjuːs/ | hipotenusa |
Which side does the ladder need?
- A ladder reaches a height of 4 m while its foot is 3 m from a wall. Which sides are known, and which angle do we need?
- This lesson studies hypotenuse 斜边: The side opposite the right angle in a right-angled triangle.
Choose the mathematical structure
- In a right triangle a²+b²=c²; sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent. For other triangles, use the sine or cosine rule, or area=ab sin C/2.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines hypotenuse? · ¿Qué descripción define correctamente la hipotenusa?
The side opposite the right angle in a right-angled triangle. · El lado opuesto al ángulo recto en un triángulo rectángulo.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
The ladder length is c=√(3²+4²)=5 m. Its angle to the ground satisfies tanθ=4/3, so θ≈53.1°. With two sides 6 and 8 enclosing 60°, c²=6²+8²-2×6×8 cos60°=52.
Right triangles and non-right triangles · Triángulos rectángulos y triángulos no rectángulos
In a right triangle a²+b²=c²; sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent · En un triángulo rectángulo a²+b²=c²; sinθ=cateto/hipotenusa, cosθ=adyacente/hipotenusa y tanθ=cateto/adyacente
Compare the model with the worked case and explain one change. · Compara el modelo con el caso resuelto y explica un cambio.
Find the hypotenuse when the legs are 3 and 4. · Encontrar la hipotenusa cuando los catetos son 3 y 4.
Pythagoras gives √(3²+4²)=5. · Pitágoras da √(3²+4²)=5.
Test a tempting shortcut
- Label sides relative to the chosen angle. Pythagoras needs a right angle. A calculator angle mode error can produce a plausible but wrong result. Keep unrounded values for later steps.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Pythagoras applies to every triangle, including triangles without a right angle. This claim is false. Explain which definition or assumption it violates.
Find sinθ when opposite=3 and hypotenuse=5. · Encontrar sinθ cuando cateto=3 e hipotenusa=5.
Sine=opposite/hypotenuse=3/5=0.6. · Seno=cateto/hipotenusa=3/5=0.6.
Pythagoras applies to every triangle, including triangles without a right angle. · Pitágoras se aplica a todo triángulo, incluidos triángulos sin ángulo recto.
Label sides relative to the chosen angle. Pythagoras needs a right angle. A calculator angle mode error can produce a plausible but wrong result. Keep unrounded values for later steps. · Etiquetar lados respecto al ángulo elegido. Pitágoras necesita un ángulo recto. Un error de modo angular en la calculadora puede producir un resultado plausible pero incorrecto. Mantener valores sin redondear para pasos posteriores.
Interpret a new situation
- Use a plan or elevation for a three-dimensional problem before applying a triangle rule. Explain why the chosen triangle contains the required length or angle.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Two sides 6 and 8 enclose 90°. Find their area. · Dos lados 6 y 8 encierran 90°. Encontrar su área.
The sides are perpendicular, so area=6×8/2=24. · Los lados son perpendiculares, así que área=6×8/2=24.
Match each part of a complete solution to its purpose. · Emparejar cada parte de una solución completa con su propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Una suposición justifica el modelo; una comprobación verifica el resultado; la interpretación lo conecta con la pregunta.
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The side opposite the right angle in a right-angled triangle. Choose the relationship, show the method, check its assumptions and interpret the result.