Question

special right triangles quiz
given that
$ab = 6$
(1) find the length of bc.
(2) find the length of ac.
given that
$ac = 24$
(3) find the length of bc.
(4) find the length of ab.
given that
$xz = 12$
(5) find the length of yz.
(6) find the length of xy.

Explanation:

Step1: Use sine for BC (AB=6)

$\sin(30^\circ) = \frac{BC}{AB} \implies BC = AB \cdot \sin(30^\circ) = 6 \cdot \frac{1}{2} = 3$

Step2: Use cosine for AC (AB=6)

$\cos(30^\circ) = \frac{AC}{AB} \implies AC = AB \cdot \cos(30^\circ) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3}$

Step3: Use tangent for BC (AC=24)

$\tan(30^\circ) = \frac{BC}{AC} \implies BC = AC \cdot \tan(30^\circ) = 24 \cdot \frac{\sqrt{3}}{3} = 8\sqrt{3}$

Step4: Use cosine for AB (AC=24)

$\cos(30^\circ) = \frac{AC}{AB} \implies AB = \frac{AC}{\cos(30^\circ)} = \frac{24}{\frac{\sqrt{3}}{2}} = 16\sqrt{3}$

Step5: Identify 45-45-90 triangle sides

In 45-45-90 triangle, legs are equal: $YZ = XZ = 12$

Step6: Use Pythagoras for XY (XZ=12)

$XY = XZ \cdot \sqrt{2} = 12\sqrt{2}$

Answer:

(1) $3$
(2) $3\sqrt{3}$
(3) $8\sqrt{3}$
(4) $16\sqrt{3}$
(5) $12$
(6) $12\sqrt{2}$