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Key Stage 4

Side lengths with Exact Values

Lesson

Worked Examples

Question 1

Find the length of side $h$h.

A right-angled triangle, $\triangle ABC$△ABC, is labeled with vertices $A$A, $B$B, and $C$C. The right angle is located at vertex $B$B indicated by a small blue square. The angle at vertex $A$A, $\angle BAC$∠BAC, measures $30^\circ$30°. The angle at vertex $C$C, $\angle BCA$∠BCA, measures $60^\circ$60° and is marked with a single blue arc. The hypotenuse $AC$AC is marked with a single tick mark. Side $AB$AB measures $\sqrt{3}$√3 units. The side $BC$BC measures $h$h units. The $60^\circ$60° angle is adjacent to side $h$h. The $30^\circ$30° angle is opposite the side $h$h.

Question 2

Consider the adjacent figure:

Solve for the unknown $\theta$θ. Leave your answer in degrees.

Question 3

Consider the adjacent figure:

A right-angled triangle is depicted with a 45-degree angle at the top. The side opposite the 45-degree angle, and also the base of the triangle, is labeled $y$y. The hypotenuse, extending from the lower right vertex to the upper left vertex, is labeled $x$x. The side adjacent to the 45-degree angle, and also the height of the triangle, is labeled with the radical expression ' $4$ (Square Root of $3$ )'. The right angle is at the lower left corner of the triangle.

Solve for the exact value of $x$x.
Solve for the exact value of $y$y.

Question 4

Consider the equilateral with side lengths of $2$2 units.

Find the perpendicular height $h$h of the triangle.
Determine the measure of angle $x$x.
Determine the exact value of $\sin60^\circ$sin60°.
Determine the exact value of $\cos60^\circ$cos60°.
Determine the exact value of $\tan60^\circ$tan60°.
Determine the exact value of $\sin30^\circ$sin30°.
Determine the exact value of $\cos30^\circ$cos30°.

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