The diagram shows a right-angled triangle.

Calculate the length of the side marked \(x\).

Give your answer to 1 decimal place.

Here is a right-angled triangle.

Give your answers correct to 1 decimal place.

The diagonal of a rectangle is \(23.5\) cm long.

The diagonal makes an angle of \(37^\circ\) with a side of length x cm.

No diagram is given.

Find the value of \(x\).

Give your answer correct to 1 decimal place.

The diagram shows a right-angled triangle.

Work out the length of the side marked \(x\).

Give your answer correct to 1 decimal place.

The diagram shows a right-angled triangle.

Calculate the exact length of the side marked \(x\).

Idris measures the ramp in a playground. It is \(14\) m long and meets the level ground at an angle of \(34^\circ\).

Give your answers correct to 1 decimal place.

Sam is finding the length of the side marked \(x\) in this right-angled triangle.

Here is Sam's working.

\(\mathrm{ tan }\,32^\circ = \frac{ 20 }{ x }\)

\(x = 20 \ ÷ \ \mathrm{ tan }^{-1}\,32^\circ\)

Here are six statements about Sam's working.

1. Sam has used the wrong ratio for the sides in the triangle. 2. Sam has put the fraction the wrong way up. 3. Sam should have used the theorem of Pythagoras, not trigonometry. 4. Sam has used the wrong length for the side that is given. 5. Sam has multiplied where it should be divided, or divided where it should be multiplied, when rearranging the equation. 6. Sam has used an inverse function to find a length.

Two of the statements are errors that Sam has made. Identify the two errors by writing down the numbers of the two statements.

The diagram shows a shape made by joining a right-angled triangle to a rectangle. The dashed line is a side of the rectangle and a side of the triangle.

Find the area of the whole shape. Give your answer correct to 3 significant figures.

The diagram shows two right-angled triangles, \(ABD\) and \(CBD\), joined along \(BD\) to make the larger triangle \(ABC\). \(ADC\) is a straight line and angle \(ADB\) is \(90^\circ\).

Find the size of angle \(BCD\).

Give your answer to 1 decimal place.

\(DCB\) is a straight line. \(AC\) is perpendicular to \(DB\), so \(ADC\) and \(ABC\) are right-angled triangles.

\(CB = 13\) cm

Angle \(ADC = 50^\circ\) and angle \(ABC = 33^\circ\).

Find the length of \(AD\).

Give your answer correct to 3 significant figures.