a triangle has sides f length xcm,(2x-1)cm and (2x+1)cm. If its perimeter is 40, state the size of the largest angle of the triangle giving reasons
P = x + 2 x - 1 + 2 x + 1 = 40
5 x = 40 Divide both sides by 5
x = 8 cm
2 x - 1 = 2 * 8 - 1 = 16 - 1 = 15 cm
2 x + 1 = 2 * 8 + 1 = 16 + 1 = 17 cm
Largest angle = 90° becouse your triangle is right triangle.
Use the Pythagorean theorem.
The square of the hypotenuse is equal to
the sum of the squares of the other two sides.
In this case:
17 ^ 2 = 15 ^ 2 + 8 ^ 2
289 = 225 + 64
So your triangle is right triangle.
To find the size of the largest angle of the triangle, we need to apply the triangle inequality theorem and the fact that the sum of the angles in a triangle is 180 degrees.
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's apply this to our triangle:
x cm + (2x - 1) cm > (2x + 1) cm
3x - 1 > 2x + 1
x > 2
Using the perimeter information, we can set up the following equation:
x cm + (2x - 1) cm + (2x + 1) cm = 40 cm
Combining like terms:
5x = 40 + 1
5x = 41
Solving for x:
x = 41/5
x = 8.2 cm
Now that we have the value of x, we can find the lengths of the sides of the triangle:
Side 1: x cm = 8.2 cm
Side 2: (2x - 1) cm = (2 * 8.2 - 1) cm = 15.4 cm
Side 3: (2x + 1) cm = (2 * 8.2 + 1) cm = 17.4 cm
To find the largest angle, we need to determine the longest side of the triangle, which is the side with the length of 17.4 cm.
Now, let's find the largest angle using the Law of Cosines:
cos(A) = (b^2 + c^2 - a^2) / (2bc)
Let's calculate the largest angle:
cos(A) = (15.4^2 + 17.4^2 - 8.2^2) / (2 * 15.4 * 17.4)
cos(A) = (237.16 + 302.76 - 67.24) / (2 * 15.4 * 17.4)
cos(A) = 472.68 / 533.04
cos(A) = 0.8864
Using an inverse cosine function, we can find the value of A:
A = acos(0.8864)
A ≈ 29.4 degrees
Therefore, the largest angle of the triangle is approximately 29.4 degrees.
To find the largest angle of the triangle, we need to first determine the values of x and the three angles of the triangle using the information provided.
We know that the perimeter of the triangle is given by the sum of the lengths of its three sides. Given that the perimeter is 40, we can write the equation:
x cm + (2x-1) cm + (2x+1) cm = 40 cm
Now, let's solve this equation to determine the value of x.
Combining like terms, we have:
5x = 40
Dividing both sides of the equation by 5, we find:
x = 8
Now that we have the value of x, we can substitute it back into the expressions for the three sides of the triangle to find their lengths:
Side 1: x cm = 8 cm
Side 2: (2x-1) cm = (2*8 - 1) cm = 15 cm
Side 3: (2x+1) cm = (2*8 + 1) cm = 17 cm
Now, let's use these side lengths to find the three angles of the triangle. To find the largest angle, we should determine the largest side and its corresponding angle.
Using the Law of Cosines, we can find the angle opposite the largest side. The formula is:
cos(A) = (b^2 + c^2 - a^2) / (2 * b * c)
Where b and c are the lengths of the two sides forming the angle A, and a is the length of the side opposite the angle A.
Let's calculate the cosines of all three angles:
cos(A) = [(15^2) + (17^2) - (8^2)] / [2 * 15 * 17] ≈ 0.976
cos(B) = [(15^2) + (8^2) - (17^2)] / [2 * 15 * 8] ≈ -0.735
cos(C) = [(17^2) + (8^2) - (15^2)] / [2 * 17 * 8] ≈ 0.482
To find the largest angle, we need to find the angle with the largest cosine value. The largest cosine value is approximately 0.976, which corresponds to angle A.
Therefore, the largest angle of the triangle is angle A.
Note: The above explanation guides you through the step-by-step process of solving the problem. However, it is always a good practice to double-check the calculations to ensure accuracy.