The measure of one of the equal angles in an isosceles triangle is twice the measure of the remaining angle. Determine the exact radian measures of the three angles in the triangle.
Let x represent the 3rd angle
Thus 2x represents the other 2 angles
We know that a triangle totals 180 degress
180 = x + 2(2x)
180 = 5x
x = 36 degrees
2x
2(36) = 72 degrees
Convert to radians using the pi/180 formula
Therefore the angles are pi/5, 2pi/5, 2pi/5
Let each of the equal angles be A.
Since we know that the angles of a triangle add up to 180°, we determine that the third angle is (180-2A)°.
Thus
A=2(180-2A)
5A = 360
A = 72
Therefore the angles are 72°, 72° and 36°.
Ah, an isosceles triangle, the hipster of triangles, always trying to be different. Alright, let's break it down:
Let's call the measure of the two equal angles in the triangle "x." According to the information given, one of these angles is twice the measure of the third angle, so let's call the third angle "y."
Now we know that the sum of all the angles in a triangle is 180 degrees (or π radians if you prefer to be fancy). So we can set up an equation:
x + x + y = 180
Since we know that one of the equal angles is twice the measure of the third angle, we can say:
2y + y = 180
Simplifying that equation, we can say:
3y = 180
And solving for the value of y, we get:
y = 60
So, one of the equal angles is twice the measure of y, which means one of the equal angles is 2 * 60 = 120 degrees (or 2π radians).
Now we just need to find the other equal angle. Since the two equal angles add up to 180 degrees (or π radians), we can say:
x + x = 180 - 120
Simplifying that equation, we get:
2x = 60
So, the other equal angle is 60 degrees (or π/3 radians).
Therefore, the exact radian measures of the three angles in the isosceles triangle are π/3, π/3, and 2π/3.
Let's denote the measure of one of the equal angles in the isosceles triangle as x. According to the problem, the measure of the remaining angle is half of x.
Since the sum of the angles in a triangle is always 180 degrees (π radians), we can set up the following equation:
x + x + (1/2)x = 180°
Combining like terms:
(2 + 1/2)x = 180°
To simplify the equation, we need to convert degrees to radians. There are π radians in 180 degrees:
(2 + 1/2)x = π radians
Multiplying both sides by 2:
4x + x = 2π radians
Simplifying:
5x = 2π radians
Dividing by 5:
x = (2π/5) radians
So, one of the equal angles in the isosceles triangle is (2π/5) radians.
To find the measure of the remaining angle, we can substitute this value back into the equation:
(1/2)(2π/5) = (π/5) radians
Therefore, the measure of the remaining angle is (π/5) radians.
Since the triangle is isosceles, the remaining angle must also be (π/5) radians.
Therefore, the exact radian measures of the three angles in the triangle are:
(2π/5) radians, (π/5) radians, and (π/5) radians.
To determine the exact radian measures of the three angles in the triangle, we can use the fact that the sum of the angles in any triangle is always equal to 180 degrees (π radians).
Let's assume that the measure of one angle in the isosceles triangle is x radians. Since the measure of one of the equal angles is twice the measure of the remaining angle, we can define the other equal angle as 2x radians.
We can now set up an equation to find the measure of the remaining angle. Since the sum of the three angles in the triangle is 180 degrees (π radians), we have:
x + 2x + remaining angle = 180 degrees (π radians)
Combining like terms, we have:
3x + remaining angle = 180 degrees (π radians)
Simplifying further, we get:
remaining angle = 180 degrees - 3x (π radians)
Now, we can substitute this expression for the remaining angle into our equation:
x + 2x + 180 degrees - 3x = 180 degrees (π radians)
Simplifying the equation, we get:
180 degrees - x = 180 degrees (π radians)
Now, we can solve for x:
x = 0 degrees (0 radians)
Substituting this value of x back into our previous expressions, we can find the measures of the three angles in the triangle:
First angle: x = 0 degrees (0 radians)
Second angle: 2x = 2(0) = 0 degrees (0 radians)
Remaining angle: 180 degrees - 3x = 180 degrees - 3(0) = 180 degrees (π radians)
Therefore, the exact radian measures of the three angles in the triangle are:
First angle: 0 radians
Second angle: 0 radians
Remaining angle: π radians