Problem 28
Question
Geometry One of the congruent sides of an isosceles triangle is 10 \(\mathrm{cm}\) long. One of the congruent angles has a measure of \(54^{\circ} .\) Find the perimeter of the triangle. Round your answer to the nearest centimeter.
Step-by-Step Solution
Verified Answer
The perimeter of the isosceles triangle is approximately the sum of both 10 cm sides and the rounded base length.
1Step 1: Understanding the Property of Isosceles Triangles
An isosceles triangle has two sides that are equal in length and two angles that are equal in measure. In this problem, we know the length of one of the congruent sides and the measurement of one of the congruent angles.
2Step 2: Using the Given Congruent Side Length to Determine the Other
Since the triangle is isosceles, the other congruent side is also 10 cm long.
3Step 3: Using the Law of Cosines to Find the Base
We can use the Law of Cosines to find the length of the base (the third side) of the triangle. The formula is \(c^2 = a^2 + b^2 - 2ab\cos(C)\), where \(a\) and \(b\) are the lengths of the congruent sides, \(C\) is the included angle, and \(c\) is the length of the base we want to calculate.
4Step 4: Calculate the Base Length
Substitute the known values into the Law of Cosines equation: \(c^2 = 10^2 + 10^2 - 2(10)(10)\cos(54^\circ)\). Then calculate the value of \(c\), which will be the length of the base.
5Step 5: Rounding the Base Length
After calculating the value of \(c\), if it is not a whole number, round it to the nearest centimeter.
6Step 6: Calculating the Perimeter
Add the lengths of all three sides to find the perimeter: one of the congruent sides (10 cm) + the other congruent side (10 cm) + the base (rounded length from previous step).
Key Concepts
Law of CosinesIsosceles Triangle PropertiesTriangle Perimeter Calculation
Law of Cosines
The Law of Cosines is an essential rule in trigonometry for calculating the lengths of sides in any triangle, not only right triangles. It serves as an extension of the Pythagorean Theorem, which applies only to right triangles. The Law of Cosines comes into play when you're dealing with non-right triangles where knowing two sides and the included angle (the angle between the two known sides) is critical.
For an isosceles triangle, the formula reads: \[ c^2 = a^2 + b^2 - 2ab\cos(C) \] where \(a\) and \(b\) are the lengths of the two equal sides, \(C\) is the measurement of the included angle, and \(c\) is the base or the side opposite the included angle. By substituting the given values into the formula, you can solve for the unknown side. This method is particularly useful when the triangle's characteristics don't meet the conditions of simpler rules, such as those that are more straightforward for right triangles.
For an isosceles triangle, the formula reads: \[ c^2 = a^2 + b^2 - 2ab\cos(C) \] where \(a\) and \(b\) are the lengths of the two equal sides, \(C\) is the measurement of the included angle, and \(c\) is the base or the side opposite the included angle. By substituting the given values into the formula, you can solve for the unknown side. This method is particularly useful when the triangle's characteristics don't meet the conditions of simpler rules, such as those that are more straightforward for right triangles.
Isosceles Triangle Properties
Isosceles triangles have unique properties that distinguish them from other types of triangles. Notably, they have at least two sides of equal length, which are called the legs, and at least two angles of equal measure, known as the base angles. These features bear significance when it comes to calculations involving isosceles triangles.
It's critical to remember that in an isosceles triangle, the angles opposite the equal sides are also equal. In solving problems, knowing just a few attributes of the triangle (like a side and an angle, as given in the exercise) allows you to determine the other parts using geometric principles and trigonometric identities. Moreover, the symmetry in isosceles triangles simplifies calculations compared to those involving scalene triangles, where all three sides and angles might differ.
It's critical to remember that in an isosceles triangle, the angles opposite the equal sides are also equal. In solving problems, knowing just a few attributes of the triangle (like a side and an angle, as given in the exercise) allows you to determine the other parts using geometric principles and trigonometric identities. Moreover, the symmetry in isosceles triangles simplifies calculations compared to those involving scalene triangles, where all three sides and angles might differ.
Triangle Perimeter Calculation
To find the perimeter of a triangle, simply add up the lengths of all its sides. For an isosceles triangle, the task can be even more straightforward, as you know that two of the sides are identical. After you find the third side (the base) using the Law of Cosines or other applicable methods, the perimeter (\(P\text)\) can be calculated with the equation \[ P = a + a + c \], where \(a\) represents the length of one of the equal sides and \(c\) is the base.
In the problem provided, after determining the length of the base, you would just double the length of one of the known sides and add the base length to it. These simple steps result in the total perimeter of the isosceles triangle. Remember, if your base length calculation doesn't yield a whole number, to round it to the closest whole number for the final perimeter calculation.
In the problem provided, after determining the length of the base, you would just double the length of one of the known sides and add the base length to it. These simple steps result in the total perimeter of the isosceles triangle. Remember, if your base length calculation doesn't yield a whole number, to round it to the closest whole number for the final perimeter calculation.
Other exercises in this chapter
Problem 28
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