Triangle ABC has area equal to (90 X spuare root 3)/4 and perimeter equal to 30. Also, one of its angles is equal to 60∘. What is the product of the sides of ABC?
Well, well, well! It seems like we have a math problem in our clown circus today, don't we? Let's dive right into it!
We know that the area of triangle ABC is given by (90 * square root 3)/4, and the perimeter is 30. Plus, one of the angles in the triangle is 60∘.
First, let's find the lengths of the sides of the triangle. Since the triangle has an angle of 60∘, we have ourselves an equilateral triangle, my friend! That means all sides are equal in length.
Now, we need to find the length of each side. To do that, we'll use the perimeter. Since the perimeter is 30, and there are three equal sides, each side of the triangle will be 30/3 = 10 units long.
So, the length of each side of triangle ABC is 10 units. Now, let's find the product of these sides: 10 * 10 * 10 = 1000.
Tada! The product of the sides of triangle ABC is 1000 units. I hope you're enjoying this clown-tastic math show!
To find the product of the sides of triangle ABC, we need to determine the lengths of all three sides first.
Let's label the sides of triangle ABC as a, b, and c. Given that the area of the triangle is (90√3)/4 and the perimeter is 30, we can use these pieces of information to solve for the side lengths.
The area of a triangle can be expressed as Area = (1/2) * base * height. In this case, we have an equilateral triangle, so the height can be determined using the formula height = (√3/2) * side length.
Substituting the values into the area formula, we have:
(90√3)/4 = (1/2) * side length * (√3/2) * side length.
Simplifying the equation, we get:
(90√3)/4 = (1/2) * (√3/2) * (side length)^2.
(90√3)/4 = (√3/4) * (side length)^2.
Canceling out the common terms, the equation becomes:
90 = (side length)^2.
Taking the square root of both sides, we find:
side length = √90.
Given that we have an equilateral triangle, all three sides are equal.
So, a = b = c = √90.
The perimeter of the triangle is the sum of all three sides, which gives us:
30 = a + b + c.
Now we substitute the value of a into the equation:
30 = √90 + √90 + √90.
Simplifying, we have:
30 = 3√90.
Dividing both sides by 3, we get:
10 = √90.
Squaring both sides of the equation, we find:
100 = 90.
This is not true, so there must be an error in our calculations.
Let's go through the steps again:
The area of a triangle can be expressed as Area = (1/2) * base * height. In this case, we have an equilateral triangle, so the height can be determined using the formula height = (√3/2) * side length.
Substituting the values into the area formula, we have:
(90√3)/4 = (1/2) * side length * (√3/2) * side length.
Simplifying the equation, we get:
(90√3)/4 = (√3/4) * (side length)^2.
Dividing both sides of the equation by (√3/4), we find:
side length^2 = (90√3)/√3.
Simplifying the right side of the equation, we have:
side length^2 = 90.
Taking the square root of both sides, we get:
side length = √90.
So, a = b = c = √90.
Now, let's substitute the values of a into the perimeter equation:
30 = √90 + √90 + √90.
Simplifying, we have:
30 = 3√90.
Dividing both sides by 3, we get:
10 = √90.
Squaring both sides of the equation, we find:
100 = 90.
This is still not true. Therefore, there must have been an error in the question or the given information. Please double-check the details, and feel free to ask any other question.
To solve this problem, we can use the given information about the area, perimeter, and angle of triangle ABC.
Let's first find the lengths of the sides of the triangle.
We know that the perimeter of the triangle is equal to 30. Since there are three sides in a triangle, let's denote the lengths of the sides as a, b, and c.
The perimeter of a triangle is the sum of its sides, so we can write the equation:
a + b + c = 30
Next, let's use the given information about the angle to find the relationship between the sides of the triangle.
If one of the angles is 60∘, it means that the triangle is a special type of triangle called an equilateral triangle. In an equilateral triangle, all three sides are equal.
Therefore, we can write the equation:
a = b = c
Now, we can substitute this equation into the perimeter equation:
a + a + a = 30
3a = 30
a = 10
Since all sides are equal, we know that the lengths of all sides of the triangle are 10.
Finally, to find the product of the sides, we can simply calculate:
Product of sides = a * b * c = 10 * 10 * 10 = 1000
Therefore, the product of the sides of triangle ABC is 1000.