The altitude to the legs is 5 centimeters. The area of the triangle is approximately 21.651 square centimeters. The perimeter of the triangle is 54.641 centimeters.
Let be both [tex]l[/tex] and [tex]\theta[/tex] the length of the legs, in centimeters, and the angle between the legs, in sexagesimal degrees. The altitude and the base of the triangle ([tex]h, b[/tex]) are, respectively:
[tex]b = 2\cdot l \cdot \sin 0.5\theta[/tex] (1)
[tex]h = l\cdot \cos 0.5\theta[/tex] (2)
The area ([tex]A[/tex]), in square centimeters, and perimeter of the triangle ([tex]p[/tex]), in centimeter are expressed by these formulas:
[tex]A = 0.5\cdot l^{2}\cdot \sin 0.5\theta \cdot \cos 0.5\theta[/tex] (3)
[tex]p = 2\cdot l + b[/tex] (4)
If we know that [tex]l = 10\,cm[/tex] and [tex]\theta = 120^{\circ}[/tex], then the altitude to the legs, the area and the perimeter of the triangle are, respectively:
Altitude to the legs
[tex]h = (10\,cm)\cdot \cos 60^{\circ}[/tex]
[tex]h = 5\,cm[/tex]
The altitude to the legs is 5 centimeters.
Area
[tex]A = 0.5\cdot (10\,cm)^{2}\cdot \sin 60^{\circ}\cdot \cos 60^{\circ}[/tex]
[tex]A \approx 21.651\,cm^{2}[/tex]
The area of the triangle is approximately 21.651 square centimeters.
Perimeter
[tex]b = 2\cdot (10\,cm)\cdot \cos 60^{\circ}[/tex]
[tex]b = 34.641\,cm[/tex]
[tex]p = 34.641\,cm + 2\cdot (10\,cm)[/tex]
[tex]p = 54.641\,cm[/tex]
The perimeter of the triangle is 54.641 centimeters.
We kindly invite to check this question on triangles: https://brainly.com/question/2269348
