Final answer:
To derive the area formula for an isosceles triangle, we bisect it to form right triangles and use trigonometric identities to express the area in terms of the length of the side and the vertex angle.
Step-by-step explanation:
To show that the area A of an isosceles triangle with equal sides of length x is A = 1/2 x² sin(θ), where θ is the angle formed by the two equal sides, we start by drawing the triangle and dropping a perpendicular from the vertex with angle θ to the base, bisecting the base and creating two right triangles.
In each right triangle, the hypotenuse is x, and angle θ/2 is adjacent to half the base. Therefore, the length of half the base (let's call it b) can be found using the cosine function: b = x cos(θ/2). Now, the area of the isosceles triangle is twice the area of one of these right triangles.
The height h of the right triangle is the opposite side to angle θ/2 and can be found using the sine function: h = x sin(θ/2). The area of the right triangle is thus 1/2 base x height = 1/2 (2b) h. Substituting the expressions for b and h, we have 1/2 (2 x cos(θ/2)) (x sin(θ/2)).
Using the double-angle identity for sine, sin(θ) = 2 sin(θ/2) cos(θ/2), we can rewrite the area as 1/2 x² sin(θ), as required.