find the perimeter of an isosceles triangle whose base is 20cm and the vertex angle is 120 degrees.
draw an altitude. That will give you two right triangles that are 30-60-90 degrees.
Since the sides of those triangles are in the ratio of 1:√3:2, and you have the long leg as half the base, or 10cm, the hypotenuse (the side of the isosceles triangle) has length 20/√3.
So, the triangle has perimeter
20 + 20√3 + 20√3
To find the perimeter of an isosceles triangle, we need to know the lengths of the two equal sides. However, in this case, only the base length is provided. So, we need to find the length of the other two equal sides.
In an isosceles triangle, the base angles are equal. Since the vertex angle is 120 degrees, the base angles will each be (180 - 120) / 2 = 60 degrees.
We can use the law of sines to find the length of the equal sides. The law of sines states that the ratio of the length of a side to the sine of the opposite angle is the same for all three sides of a triangle.
Let's assume the length of each equal side is "x".
So, we have:
sin(60) / x = sin(120) / 20
Using the trigonometric identity sin(120) = sin(180 - 120) = sin(60), we can simplify the equation to:
sin(60) / x = sin(60) / 20
Cross-multiplying, we get:
20 * sin(60) = x * sin(60)
sin(60) cancels out, leaving us with:
x = 20
Therefore, the length of each equal side is 20 cm.
Now, we can calculate the perimeter of the triangle:
Perimeter = Length of Base + 2 * Length of Equal Sides
Perimeter = 20 + 2 * 20
Perimeter = 20 + 40
Perimeter = 60 cm
So, the perimeter of the isosceles triangle is 60 cm.
To find the perimeter of an isosceles triangle, we need to know the lengths of its sides. However, in this case, we only have the length of the base and the measure of the vertex angle.
An isosceles triangle has two equal sides and two equal angles opposite those sides. In this case, the vertex angle is 120 degrees, so the other two angles must both be (180 - 120) / 2 = 30 degrees each.
To find the length of the equal sides, we can use the trigonometric concept of the sine function. The sine function relates the ratio of the length of a side to the length of the hypotenuse (the longest side) to the sine of the opposite angle.
Let's denote the length of the equal sides as "a". We can form a right triangle by drawing a perpendicular from the vertex angle to the base of the triangle. This right triangle will have one leg of length a/2 (half of an equal side) and the opposite angle of 30 degrees.
Using the sine function, we have sin(30 degrees) = (a/2) / 20cm. Rearranging this equation, we can solve for "a":
a/2 = 20cm * sin(30 degrees)
a/2 = 20cm * (1/2)
a/2 = 10cm
Multiplying both sides by 2 gives us:
a = 2 * 10cm
a = 20cm
Now that we know the length of the equal sides is 20cm, we can calculate the perimeter of the isosceles triangle by adding up the lengths of all three sides:
Perimeter = 20cm + 20cm + 20cm
Perimeter = 60cm
Therefore, the perimeter of the given isosceles triangle is 60cm.