One of the equal angles of an isosceles triangle is 55 degrees. If the base is 20 cm, what is the perimeter and the area?
So the other angle must be 180-2(55) or 70°
draw in the height, h
You now have a right-angled triangle, let the hypotenuse be x
you can find the height by
tan 55° = h/10
you can find the side by
cos 55° = 10/x
once you have x, you can find the perimeter.
once you have h, you can find the area by
area = (1/2)(20)(h)
dvvdb
To find the perimeter of an isosceles triangle, we need to know the length of all three sides. Let's start by finding the missing angle.
In an isosceles triangle, the two equal angles are opposite the two equal sides. Since one of the equal angles is given as 55 degrees, the other equal angle will also be 55 degrees.
To find the third angle, we can use the fact that the sum of the angles in a triangle is always 180 degrees. Since we know two of the angles (55 degrees and 55 degrees), we can subtract their sum from 180 degrees to find the third angle:
Third angle = 180 degrees - 55 degrees - 55 degrees
Third angle = 70 degrees
Now that we know all three angles, we can use the Law of Sines to find the lengths of the sides. However, in order to use the Law of Sines, we need to know at least one side length and its opposite angle.
In this case, the base of the triangle is given as 20 cm. Let's label the two equal sides of the triangle as "a" and the base as "b". Since the two equal angles (55 degrees) are opposite to the two equal sides, we know that "a" is equal to "a".
Now we can use the Law of Sines, which states that the ratio of the length of a side to the sine of its opposite angle is constant:
a/sin(55) = b/sin(70)
We can rearrange this equation to solve for "a":
a = b * sin(55) / sin(70)
Since we know the value of "b" (20 cm), we can substitute it into the equation:
a = 20 * sin(55) / sin(70)
Using a calculator, we can find that a ≈ 16.32 cm.
Now we know the lengths of all three sides: "a" is approximately 16.32 cm, "a" is approximately 16.32 cm, and "b" is 20 cm.
To find the perimeter, we simply add up the lengths of all three sides:
Perimeter = a + a + b
Perimeter = 16.32 cm + 16.32 cm + 20 cm
Perimeter ≈ 52.64 cm
So, the perimeter of the isosceles triangle is approximately 52.64 cm.
To find the area of the triangle, we can use the equation:
Area = (1/2) * base * height
The base is given as 20 cm. To find the height, we can use the Pythagorean theorem.
Since the triangle is isosceles, we can divide it into two congruent right-angled triangles. The base of each right-angled triangle is half of the base of the isosceles triangle, which is 20 cm divided by 2, or 10 cm.
Now, let's label the height of each right-angled triangle as "h". We can use the Pythagorean theorem to find the height:
h^2 = a^2 - (b/2)^2
h^2 = 16.32 cm^2 - (10 cm)^2
h^2 = 266.4624 cm^2 - 100 cm^2
h^2 = 166.4624 cm^2
h ≈ √166.4624 cm^2
h ≈ 12.9 cm
Now that we know the height, we can calculate the area of the isosceles triangle:
Area = (1/2) * base * height
Area = (1/2) * 20 cm * 12.9 cm
Area ≈ 129 cm^2
So, the area of the isosceles triangle is approximately 129 cm^2.