# Triangle ABC has area 25*sqrt(3). if Angle BAC=30 degrees, find |AC|=|BC|=?

• MHB
• Elissa89
In summary, the triangle ABC has an area of 25*sqrt(3) and an angle BAC of 30 degrees. To find the side lengths |AC| and |BC|, the triangle is assumed to be isosceles with an angle C of 120 degrees and side lengths of a = BC = b = AC. Using the formula for area, we can solve for a and get a value of 10*4th root(3). This answer may be correct, but it is recommended to double check with your professor as |AC| = |BC| may not be a notation they use.
Elissa89
Triangle ABC has area 25*sqrt(3). if Angle BAC=30 degrees, find |AC|=|BC|=?

the answer I got was 10*4th root(3)

Is this correct?

I am asking because someone other than my professor wrote the study guid for us for the final and I am not 100% sure what |AC|=|BC| means as my professor never used it. I'm assuming it means side lengths of the triangle.

Elissa89 said:
Triangle ABC has area 25*sqrt(3). if Angle BAC=30 degrees, find |AC|=|BC|=?

the answer I got was 10*4th root(3)

Is this correct?

I am asking because someone other than my professor wrote the study guid for us for the final and I am not 100% sure what |AC|=|BC| means as my professor never used it. I'm assuming it means side lengths of the triangle.

the triangle is isosceles with $m\angle C = 120^\circ$ and $a = BC = b = AC$

$Area = \dfrac{1}{2}ab\sin(C)$

$25\sqrt{3} = \dfrac{1}{2}a^2 \sin(120^\circ)$

try again to solve for $a$

## 1. How do you find the length of sides AC and BC in Triangle ABC?

To find the length of sides AC and BC, we need to use the formula for the area of a triangle: A = 1/2 * base * height. In this case, the base is the length of side AC or BC, and the height is the distance from the opposite vertex to the base. So, we can set up the equation: 25*sqrt(3) = 1/2 * |AC| * |BC| * sin(30). Solving for |AC| and |BC|, we get |AC| = |BC| = 10 units.

## 2. Why do we use the sine function in the formula for the area of a triangle?

The sine function is used because it represents the ratio of the opposite side to the hypotenuse in a right triangle. In this case, we are finding the area of a triangle with a 30-degree angle, which means we are dealing with a special type of right triangle called a 30-60-90 triangle. The sine of 30 degrees is equal to 1/2, which is why it appears in the formula.

## 3. Can we use any other trigonometric function to solve this problem?

Yes, we can also use the cosine function in the formula for the area of a triangle. In this case, the equation would be: 25*sqrt(3) = 1/2 * |AC| * |BC| * cos(60). Solving for |AC| and |BC|, we would get the same result of |AC| = |BC| = 10 units.

## 4. Is the length of sides AC and BC always equal in a triangle with a 30-degree angle and area 25*sqrt(3)?

Yes, in this specific scenario, the length of sides AC and BC will always be equal. This is because the area of the triangle is determined by the length of the sides and the angle between them. In a 30-60-90 triangle, the sides are in the ratio of 1:sqrt(3):2, which means that the length of sides AC and BC will always be equal.

## 5. How can we prove that the length of sides AC and BC are equal in this triangle?

One way to prove that the length of sides AC and BC are equal is by using the Pythagorean Theorem. In a 30-60-90 triangle, the length of the hypotenuse (in this case, side AB) is equal to twice the length of the shorter leg (side AC). So, we can set up the equation: (2*|AC|)^2 = |AB|^2. Similarly, for side BC, we have: (|BC|)^2 = |AB|^2. Since the lengths of AB in both equations are equal, we can equate the two equations and get (2*|AC|)^2 = (|BC|)^2. Simplifying, we get |AC| = |BC| = 10 units.

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