Find the area of each triangle ABC.

A = 59.80...

Question

Find the area of each triangle ABC.

A = 59.80°, b = 15.00 cm, C = 53.10°

Accepted Answer

Hello, today we are going to determine the area of a triangle. ABC. With the following measurements, we are given that angle A is equal to 62.34 degrees. Side length B is equal to 23 centimeters and angle C is equal to 42.87 degrees. Let's go ahead and label all this information on a generic triangle. So for the triangle, we have side length, A side length B and side length C si side length A will have a respective angle. A side length B will have the respective angle B and side length C will have a respective angle C. We are told that A or angle A is equal to 62.34 degrees. So let's go ahead and label that we are told that S length B is equal to 23 centimeters. And we are told that angle C is equal to 42.87 degrees. So how can we use this information to find the area of a triangle? Well, normally if you are given two S length measurements and at least one angle of the triangle, there is a special form of the area of a triangle. You can use, there are three forms of a total. The area of the triangle is equal to one half multiplied by si length, A multiplied by S length C multiplied by sine of angle B. You can also choose to use one half multiplied by side length, A multiplied by side length B multiplied by sine of angle C. And you may also choose to use one half multiplied by S length B multiplied by S length C multiplied by sine of angle A, these, these are the three forms of the area of a triangle. You can use if you are given at least two side links and one angle of the triangle. But currently, we have two angles and one side length. What we can do is we can use the, the, some property and the law of signs to help us get the information, we need to find the area of the triangle. We can solve for angle B by using the sum property of the angles of the triangle. Now that states that the sum of the three angles of a triangle must equal to 180 degrees. So first we're going to solve for angle B, we are going to add 42.87 degrees plus 62.34 degrees plus angle B. And the sum of these three angles should equal to 180 degrees. So let's go ahead and solve for B 42.87 plus 62.34 will give us 105.21 plus the angle B is equal to 180 degrees. Subtracting 105.21 to both sides of the equation will give us an angle B is equal to 74.79 degrees. So we know that angle B is 74.79 degrees. Now, what we are going to do is we're going to use the law of signs to solve for either side length A or side length C. See the reason for this is because we now have the angle B and the side length B, which is what we need for one of the two E, one of the three equations. But we can now choose to solve for either side length A or side length C in order to solve for the areas. So let's go ahead and use the law of science to solve for side length A. Now the law of signed states that we have three equivalent ratios, we have the sine of angle A divided by sine length A is equal to the sine of angle B divided by the side length B is equal to the sine of side length C or angle C divided by the side length C. Since we have all the measurements for B and we want to solve for the side length A, we're going to take the ratio with respect to B and set it equal to the ratio with respect to A. That will leave us with a sign of A divided by silence A is equal to sine of angle B divided by sin length B. In order to solve for A, we're going to cross multiply the two ratios. This will leave us with side length A multiplied by sine of B is equal to sin length B multiplied by sine of A. In order to solve for A, we will divide both sides of this equation by sine of B. This will leave us with si length A is equal to SI length B multiplied by sine of A divided by sine of B. Now we know that the side length of B is 23 centimeters, we know the angle A is 62.34 degrees. And we solve for the angle B which is 74.79 degrees. Plugging in these values will give us sign length A is equal to 23 centimeters multiplied by sign of 62.34 degrees divided by s of 74.79 degrees. At this point. I would recommend plugging in this value into a calculator to help you solve. This will give you the value of side length A is equal to 21.11 ft. So we just solve for secondary side length, we know that the side length B is given to us as 23.00 centimeters. And we can use the angle C which was given to us as 42.87 degrees. We're going to use the equation of the area where A is equal to one half multiplied by sin length. A multiplied by S length B multiplied by sine of angle C plugging in the values will give us one half multiplied by 21.11 multiplied by 23 0.00 multiplied by sign of 42.87 multiplying 21.11 23.00 and a half together will leave us with 242.77 multiplied by sign of 42.87. And using a calculator multiplying both of these values together will give us a final value of 165.16 ft squared. This is going to be the area of the given ABC triangle. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Find the area of each triangle ABC.

A = 59.80°, b = 15.00 cm, C = 53.10°

Key Concepts

Law of Sines

Triangle Angle Sum Property

Area Formula Using Two Sides and Included Angle

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