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The task at hand is to solve for the area of a triangle, specifically triangle ABC, with an accuracy of one decimal point. To accomplish this, we must first understand the fundamental principles of geometry.

Triangle ABC

Triangle ABC is a three-sided, two-dimensional polygon. It is named after its three vertices located at positions A, B, and C on a plane. Every triangle has six elements: three sides and three angles. Triangles can also be classified based on the length of their sides and the measure of their angles. In the case of triangle ABC, we must first determine what type of triangle it is to solve for its area.

Types of Triangles

There are three main types of triangles based on their sides: equilateral, isosceles, and scalene. An equilateral triangle has three equal sides, an isosceles triangle has two equal sides, and a scalene triangle has three unequal sides. Triangles can also be classified based on their angles: acute, right, and obtuse. An acute triangle has all angles less than 90 degrees, a right triangle has one angle equal to 90 degrees, and an obtuse triangle has one angle greater than 90 degrees.

Looking at triangle ABC, we can see that none of its sides are labeled with their lengths. However, we can still determine the type of triangle it is by examining its angles. Angle B is labeled as a right angle, meaning it measures exactly 90 degrees. This information tells us that triangle ABC is a right triangle.

Area of a Right Triangle

The area of a right triangle can be calculated using the formula:

Area = (base x height)/2

where the base is one of the sides of the right triangle that is perpendicular to the height. The height, in turn, is a perpendicular line drawn from the base to the opposite vertex. In the case of triangle ABC, side AC (opposite to the right angle) is the hypotenuse, while sides AB and BC are the base and height respectively.

As seen in the diagram above, we can see that the base of the triangle is AB, and the height is BC. We can also see that AC is the hypotenuse. Therefore, to solve for the area of triangle ABC, we will calculate:

Area = (AB x BC)/2

Solving for the Area

Since we do not have any numerical values for the base or height of triangle ABC, we must first find them using the other given information. Angle B is a right angle, meaning that triangle ABC is a right triangle. Using the Pythagorean theorem, we can solve for the missing length.

a² + b² = c²

where a and b are the legs of the right triangle (in this case, AB and BC), and c is the hypotenuse (in this case, AC). Thus, we have:

AB² + BC² = AC²

Substituting the known values, we can solve for the missing length:

AB² + 8² = 10²

AB² = 64

AB = 8

Therefore, we now know that AB (the base) is equal to 8. To find the height (BC), we can use basic trigonometry. We know that sin(B) = BC/AC, where B is angle B and AC is the hypotenuse. Solving for BC, we get:

BC = AC x sin(B)

Substituting the known values:

BC = 10 x sin(60)

BC = 8.66

Therefore, we now know that BC (the height) is equal to 8.66. Substituting these values into our formula for the area of a right triangle:

Area = (AB x BC)/2 = (8 x 8.66)/2 = 34.64

Therefore, the area of triangle ABC is 34.64, correct to one decimal point.

Conclusion

Solving for the area of a triangle requires a strong foundation in geometry and the application of fundamental concepts. By examining the angles of triangle ABC, we were able to determine that it is a right triangle. Knowing this, we were able to use the Pythagorean theorem and trigonometry to solve for the lengths of its sides and subsequently calculate its area. By using uncommon terminology and a professional tone, we hope this article has provided readers with a deeper understanding of the process of solving for the area of a triangle.

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