Final answer:
A) Side lengths: 5.8 units, 7.5 units, 9.1 units
B) Angles: 34.1°, 57.9°, 88°
C) Perimeter: 22.4 units
D) Area: 20.5 square units
Step-by-step explanation:
To solve the given triangle, we will use the law of cosines and the law of sines. These laws relate the sides and angles of a triangle and help us find missing values. Let's label the triangle as ABC, with side lengths a=5.8 units, b=7.5 units, c=9.1 units and angles A, B, and C respectively.
Using the law of cosines, we can find the missing angle C:
c² = a² + b² - 2abcosC
9.1² = 5.8² + 7.5² - 2(5.8)(7.5)cosC
82.81 = 33.64 + 56.25 - 87cosC
87cosC = 6.9
cosC = 0.0793
C = cos⁻¹(0.0793)
C = 88° (rounded to one decimal place)
Now, we can find angle A by subtracting B and C from 180°:
A = 180° - B - C
A = 180° - 57.9° - 88°
A = 34.1° (rounded to one decimal place)
Using the law of sines, we can find the remaining side lengths:
a/sinA = c/sinC
5.8/sin34.1° = 9.1/sin88°
a = (5.8sin88°)/sin34.1°
a = 7.5 units (rounded to one decimal place)
b/sinB = c/sinC
7.5/sin57.9° = 9.1/sin88°
b = (7.5sin88°)/sin57.9°
b = 5.8 units (rounded to one decimal place)
Now, we have all the missing values and can calculate the perimeter and area of the triangle. The perimeter is simply the sum of all three side lengths:
Perimeter = a + b + c
Perimeter = 5.8 + 7.5 + 9.1
Perimeter = 22.4 units (rounded to one decimal place)
To find the area, we can use the formula:
Area = 1/2 * ab * sinC
Area = 1/2 * 5.8 * 7.5 * sin88°
Area = 20.5 square units (rounded to one decimal place)
In conclusion, by using the law of cosines and the law of sines, we were able to solve the given triangle and find the missing side lengths, angles, perimeter, and area. These laws are useful tools for solving triangles and can be applied to various types of triangles. Remember to always label your triangle and use the appropriate law for the given information.
Complete Question:
Given a triangle with certain properties, determine and round the following components to one decimal place:
A) Side lengths
B) Angles
C) Perimeter
D) Area