Question

Find the missing information for the triangle.

*not drawn to scale
• Make sure to find the missing angle measure and the 2 missing side
lengths.

Answer 1

missing angle:

180° - 90° - 30°

180° - 120°

60°

missing sides:

(a)

`\rightarrow \sf tan(x)= (opposite)/(adjacent)`

`\rightarrow \sf tan(30)= (4)/(adjacent)`

`\rightarrow \sf adjacent= (4)/(tan(30))`

`\rightarrow \sf adjacent= 4√(3)`

`\rightarrow \sf adjacent= 6.93 \ cm`

(b)

`\sf \rightarrow sin(x)= (opposite)/(hypotensue)`

`\sf \rightarrow sin(30)= (4)/(hypotensue)`

`\sf \rightarrow hypotensue= (4)/( sin(30))`

`\sf \rightarrow hypotensue= 8 \ cm`

Answer 2

m∠X = 60°

BX = 8 cm

BM = 4√3 cm

Explanation:

The sum of the interior angles of a triangle is 180°

Given:

• m∠B = 30°
• m∠M = 90°

⇒ m∠B + m∠M + m∠X = 180°

⇒ 30° + 90° + m∠X = 180°

⇒ 120° + m∠X = 180°

⇒ m∠X = 180° - 120°

⇒ m∠X = 60°

Using the sine rule to find the side lengths:

`(a)/(\sin A)=(b)/(\sin B)=(c)/(\sin C)`

(where A, B and C are the angles, and a, b and c are the sides opposites the angles)

Given:

• m∠X = 60°
• m∠B = 30°
• m∠M = 90°
• MX = 4 cm

`\implies (4)/(\sin 30\textdegree)=(BX)/(\sin 90\textdegree)=(BM)/(\sin 60\textdegree)`

`\implies BX=\sin 90\textdegree \cdot(4)/(\sin 30\textdegree)`

`=1 \cdot (4)/(\frac12)`

`=1 \cdot 4 \cdot 2`

`=8 \textsf{ cm}`

`\implies BM=\sin 60\textdegree \cdot(4)/(\sin 30\textdegree)`

`=(√(3))/(2)\cdot (4)/(\frac12)`

`=(√(3))/(2)\cdot 4 \cdot 2`

`=4√(3) \textsf{ cm}`