Question

Calculate all the unknown sides and angles of the triangle shown below, given the following measurements. Determine which statement is correct:
B=60°, a=60
Drawing not necessarily to scale.
See attached image.

Answer 1

Answer:

∠ A = 30° , b = 60$\sqrt{3}$ , c = 120

Step-by-step explanation:

The sum of the 3 angles in a triangle is 180° , that is

∠ A + ∠ B + 90° = 180° ( substitute ∠ B = 60° )

∠ A + 60° + 90° = 180°

∠ A + 150° = 180° ( subtract 150° from both sides )

∠ A = 30°

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To find b and c, given a = 60

using the tangent and cosine ratios with  the exact values

tan60° = $\sqrt{3}$ , cos60° = $\frac{1}{2}$

tan B = tan60° = $\frac{opposite}{adjacent}$ = $\frac{b}{a}$ = $\frac{b}{60}$ = $\sqrt{3}$ ( multiply both sides by 60)

b = 60$\sqrt{3}$

cos B = cos60° = $\frac{adjacent}{hypotenuse}$ = $\frac{a}{c}$ = $\frac{60}{c}$ = $\frac{1}{2}$ ( cross multiply )

c = 120

Then

∠ A = 30° , b = 60$\sqrt{3}$ , c = 120

Answer 2

Answer:

Correct statement:  c = 120

Step-by-step explanation:

The given triangle is a right triangle because the angle at vertex C opposite side c measures 90°, denoted by the small square symbol.

The interior angles of a triangle sum to 180°, so:

$A + B + C = 180^{\circ}$

Given that B = 60° and C = 90°, then:

$A + 60^{\circ} + 90^{\circ} = 180^{\circ}\\\\A + 150^{\circ} = 180^{\circ}\\\\A = 180^{\circ} - 150^{\circ}\\\\A = 30^{\circ}$

Therefore, the angles of triangle ABC are:

• A = 30°
• B = 60°
• C = 90°

This means that the triangle is a 30-60-90 triangle.

A 30-60-90 triangle is a special right triangle where the angles measure 30°, 60°, and 90°, with side lengths in the ratio 1 : √3 : 2 respectively.

Therefore, the formula for the ratio of the sides is x : x√3 : 2x, where:

• x is the shortest side opposite the 30° angle.
• x√3 is the side opposite the 60° angle.
• 2x is the longest side (hypotenuse) opposite the right angle.

Given that the side a opposite the 30° angle measures 60 units, then:

$a=60\\\\x = 60$

As side b is opposite the 60° angle, then:

$b=x\sqrt{3}\\\\b=60\sqrt{3}$

Finally, the hypotenuse c is:

$c=2x\\\\c = 2 \times 60\\\\c=120$

Therefore, the side lengths of triangle ABC are:

• a = 60
• b = 60√3
• c = 120

So, the only correct statement is:

$\LARGE\boxed{\boxed{c=120}}$