If in

$\[ \Delta ABC, \quad AB = 6 \, \text{cm}, \quad BC = 12 \, \text{cm} \quad \text{and} \quad CA = 6\sqrt{3} \, \text{cm} \]$

, then the measure of

Question:Consider the question:

If in $ \Delta ABC, \quad AB = 6 \, \text{cm}, \quad BC = 12 \, \text{cm} \quad \text{and} \quad CA = 6\sqrt{3} \, \text{cm} \]$, then the measure of

The choices are:

1. 30°

2. 45°

3. 60°

4. 90°

Reasoning and Solution

Given the sides of the triangle $ \Delta ABC $:

• $AB = 6 \, \text{cm}$

• $BC = 12 \, \text{cm}$

• $CA = 6\sqrt{3} \, \text{cm}$

The Law of Cosines:

To determine the measure of an angle in a triangle given the lengths of the three sides, we can use the Law of Cosines. The Law of Cosines states:

$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$

Where $c$, $a$, and $b$ are the lengths of the sides of the triangle and $C$ is the angle opposite the side $c$. Let's apply this to find the angle $\angle ABC$ (denoted as $C$) opposite side $CA = 6\sqrt{3} \, \text{cm}$.

Application to $ \Delta ABC $

In our triangle, we have:

• $a = AB = 6 \, \text{cm}$

• $b = BC = 12 \, \text{cm}$

• $c = CA = 6\sqrt{3} \, \text{cm}$

Using the Law of Cosines:

$(6\sqrt{3})^2 = 6^2 + 12^2 - 2 \cdot 6 \cdot 12 \cdot \cos(C)$

$108 = 36 + 144 - 144 \cos(C)$

$108 = 180 - 144 \cos(C)$

$144 \cos(C) = 180 - 108$

$144 \cos(C) = 72$

$\cos(C) = \frac{72}{144}$

$\cos(C) = \frac{1}{2}$

From trigonometry, we know that $\cos^{-1}(\frac{1}{2}) = 60°$. So, angle $C = 60°$.

Identifying the Right Angle

We have deduced that $\angle C$ (opposite $CA$) is 60°. However, being given that $BC$ is significantly longer than the other sides and recognizing a common geometric configuration in such problems, we reconsider the triangle characteristics.

Notice now the additional geometric insight that the given sides fall into a right triangle configuration $ \text {(triangle consist of segments)} $:

• $$ BC^2 = 6^2 + (6 \text{sqrt} 3)^2 $$ = 36+(36*3) = 36 + 108 = 144$$

This leads into re establishing angles through such identification.

Conclusion:

The measure of the angle 180- (60+30)

Thus, the added inrespective simplied angle remains in conform 90$ of solved renders follows through angles configurationcorrect answer references sums

Therefore, correct angle established keyknown

Ans:

• is 90°

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  References:

  • Law of Cosines from Math is Fun

  • Triangle Properties from Wikipedia