1. One angle of a rhombus measures 60 degrees. If the perimeter ois 20 inches, find the length of both diagonals.
2. An equilateral triangle has an altitude of 18 inches. Find the are and perimeter of the triangle
To solve the first problem, we can start by finding the length of one side of the rhombus. Since opposite angles in a rhombus are equal, we know that the other three angles are also 60 degrees each.
The perimeter of a rhombus is equal to four times the length of one side. Therefore, if the perimeter is 20 inches, each side measures 20 / 4 = 5 inches.
To find the length of the diagonals, we need to use the properties of a rhombus. The diagonals of a rhombus bisect each other at a right angle, creating four congruent right triangles. Each of these right triangles has one leg equal to half the length of the diagonals, and another leg equal to half the length of the sides.
In this case, since the angle in one of these right triangles is 60 degrees, we can use the trigonometric function cosine to find the length of the diagonals.
Let's call the length of half of one diagonal "d". Then, using the cosine function, we have cos(60) = (5 / 2) / d. Rearranging the equation, we get d = (5 / 2) / cos(60).
Now, we can calculate the value of cos(60) using a scientific calculator or by referring to a trigonometric table. The value of cos(60) is 0.5.
Substituting this value into the equation, we have d = (5 / 2) / 0.5 = 5 inches. Therefore, the length of each diagonal is 2 * 5 = 10 inches.
To solve the second problem, we need to find the area and perimeter of an equilateral triangle with an altitude of 18 inches.
Since it is an equilateral triangle, all three sides are equal. Let's call the length of one side "s".
To find the area of an equilateral triangle, we can use the formula A = (sqrt(3) / 4) * s^2, where sqrt(3) represents the square root of 3.
Since the altitude of the triangle bisects the base, it creates a right triangle. One leg of this right triangle is the altitude (18 inches), and the hypotenuse is the length of one side (s). Using the Pythagorean theorem, we can find the length of one side.
The Pythagorean theorem states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. In this case, 18^2 + (s / 2)^2 = s^2.
Simplifying the equation, we have 324 + (s^2 / 4) = s^2. Rearranging the equation, we get (3s^2 - 1296) / 4 = 0.
To solve this quadratic equation, we can multiply both sides by 4 to eliminate the fraction, resulting in 3s^2 - 1296 = 0.
By solving this equation, we find that s = 24 inches.
Now we can substitute this value of s into the formula for the area A = (sqrt(3) / 4) * s^2 to find the area of the equilateral triangle.
Plugging in the value of s, we have A = (sqrt(3) / 4) * 24^2 = (sqrt(3) / 4) * 576.
Using a calculator, we find that the area of the equilateral triangle is approximately 216 * sqrt(3) square inches.
To find the perimeter, we can simply multiply the length of one side by 3, since all three sides are equal. The perimeter is therefore 24 * 3 = 72 inches.