Selecting the right method

1. Legs 7 and 10 are known. Which method finds the third side? Write its equation.

   2 marks
2. Hypotenuse and an acute angle are known. Which equation finds adjacent?

   2 marks
3. Opposite and hypotenuse are known. Write the expression for the acute angle.

   2 marks
4. A right triangle has one acute angle ${45}^{\circ }$ but no side length. Can you find its hypotenuse? Explain.

   3 marks
5. A student uses $c=a+b$ for legs 3 and 4. Correct the calculation.

   2 marks
6. A right triangle has legs 3 cm and 4 cm. Find the hypotenuse, then the acute angle opposite 3 cm to two decimal places.

   4 marks

Equivalent correct methods earn the corresponding marks. Include units where asked.

1. Legs 7 and 10 are known. Which method finds the third side? Write its equation.

   2 marks

   Two sides are known; a side is missing.

   Pythagoras: $c^2=7^2+{10}^2$
2. Hypotenuse and an acute angle are known. Which equation finds adjacent?

   2 marks

   Cosine connects adjacent and hypotenuse.

   $a=hcos\theta$
3. Opposite and hypotenuse are known. Write the expression for the acute angle.

   2 marks

   Calculate $\frac{\text{opposite}}{\text{hypotenuse}}$, then use inverse sine.

   $\theta ={sin}^{-1}\left(\frac{o}{h}\right)$
4. A right triangle has one acute angle ${45}^{\circ }$ but no side length. Can you find its hypotenuse? Explain.

   3 marks

   Angles fix shape, not scale.

   Many similar triangles satisfy the angle condition.

   No
5. A student uses $c=a+b$ for legs 3 and 4. Correct the calculation.

   2 marks

   $c^2=3^2+4^2=25$.

   $c=5$, not 7.

   $c=5$
6. A right triangle has legs 3 cm and 4 cm. Find the hypotenuse, then the acute angle opposite 3 cm to two decimal places.

   4 marks

   $c=\sqrt{9+16}=5$ cm.

   $\theta ={tan}^{-1}\left(\frac{3}{4}\right)\approx {36.87}^{\circ }$.

   5 cm; ${36.87}^{\circ }$