Data: á = 32 degrees, side a = 2.5 feet and side b = 4 feet
1. Identify the swinging side and the fixed side in a possible triangle.
2. Compute the length of the shortest swinging side that will construct a triangle containing the data. (compute the length of h)
3. How many triangles are formed containing the given data? Explain your answer.
4. Sketch all possible triangles formed.
5. Use the Law of Sines to solve one of the triangles formed containing the data.
1. In this case, the swinging side refers to the side that can vary in length, while the fixed side remains constant. In this scenario, side a = 2.5 feet and side b = 4 feet are given. Therefore, the swinging side is side c.
2. To compute the length of the shortest swinging side (side c) that will construct a triangle containing the given data, we can use the Pythagorean theorem or trigonometric ratios. However, since the angle measures are not provided, we need to find one of the angles first.
To find the angle á, we can use the inverse trigonometric function, specifically arctan.
tan(á) = opposite/adjacent
tan(á) = 2.5/4
á = arctan(2.5/4)
Using a calculator, the value of á is approximately 33.75 degrees.
Now, with one angle and two sides given, we can use the sine function to find the shortest swinging side (side c).
sin(A) = opposite/hypotenuse
sin(á) = side c/4
side c = 4 * sin(á)
Using a calculator, the length of side c is approximately 2.29 feet.
3. Only one triangle can be formed containing the given data. This is because fixing two sides of a triangle uniquely determines its shape. In this case, fixing side a = 2.5 feet, side b = 4 feet, and angle á = 32 degrees determines a unique triangle.
4. To sketch the possible triangles, we can illustrate side a = 2.5 feet vertically, side b = 4 feet horizontally, and connect them with side c. However, since only one triangle can be formed, there is only one possible sketch.
5. To use the Law of Sines to solve one of the triangles formed, we can use the following formula:
a/sin(A) = b/sin(B) = c/sin(C)
Using the given values, we have:
side a = 2.5 feet (opposite angle A)
side b = 4 feet (opposite angle B)
side c = 2.29 feet (opposite angle C)
We have the values for sides a, b, and c, and angle A = 32 degrees. We can use the Law of Sines to find angle B.
a/sin(A) = b/sin(B)
2.5/sin(32) = 4/sin(B)
sin(B) = (4 * sin(32))/2.5
Using a calculator, the value of sin(B) is approximately 0.853.
To find angle B, we can use the inverse sine function, arcsin.
B = arcsin(0.853)
Using a calculator, the value of angle B is approximately 57.19 degrees.
1. To identify the swinging side and the fixed side, we need to understand the concept of a triangle. A triangle is a polygon with three sides and three angles. One of the sides is always opposite to the largest angle, which is known as the swinging side. The other two sides are fixed sides.
2. To compute the length of the shortest swinging side (h), we will use the Law of Sines, given the data provided. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
In this case, we know that side b is opposite to angle á. So, we can use the Law of Sines to calculate the length of side b:
sin(á) / b = sin(90°) / h
Since sin(90°) is equal to 1, we can simplify the equation:
sin(á) / b = 1 / h
Now, we can substitute the given values into the equation:
sin(32°) / 4 = 1 / h
We can then rearrange the equation to solve for h:
h = 4 * (1 / sin(32°))
Using a calculator, compute sin(32°) and then find the reciprocal of the result. Multiply this by 4 to find the length of the shortest swinging side (h).
3. To determine the number of triangles formed containing the given data, we can use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we have side a = 2.5 feet and side b = 4 feet. To form a valid triangle, the sum of any two sides must be greater than the third side.
So, we can check if the Triangle Inequality Theorem holds for the given data:
a + b > c
2.5 + 4 > h
6.5 > h
From the previous calculation, we determined that the length of the shortest swinging side (h) is less than 6.5 feet, so the Triangle Inequality Theorem holds.
Therefore, there is exactly one triangle that can be formed with the given data.
4. To sketch all possible triangles formed, we can start by drawing a line segment of length 2.5 feet. This will represent side a. Then, we can draw another line segment from the endpoint of side a, making an angle of 32 degrees, and with a length of 4 feet, representing side b.
Finally, we can draw line segments from the starting point of side a and the endpoint of side b, connecting them to form side c (the swinging side). The length of side c will vary depending on the calculations from Step 2, but it should satisfy the Triangle Inequality Theorem.
By varying the length of side c within the valid range, we can sketch all possible triangles that can be formed with the given data.
5. To solve one of the triangles formed containing the given data, we can use the Law of Sines again.
Let's select the triangle where side c (the swinging side) has the length of h that we computed earlier.
Using the Law of Sines:
sin(á) / a = sin(ß) / b = sin(γ) / c
We know that side c is the swinging side with length h, á = 32 degrees, side a = 2.5 feet, and side b = 4 feet. Plugging these values into the equation, we can solve for the remaining angles or sides, depending on what the question requires.