show that the area(A) of an isosceles triangle, whose equal sides are of length s and angle between them is theta, is A=1/2s^2sin theta
You know the althtude is S*sinTheta from the diagram
Bsase is s
Area=1/2 base*altitude.
To prove that the area (A) of an isosceles triangle, with equal sides of length s and angle between them as theta, is given by A = (1/2) * s^2 * sin(theta), we can use basic geometry and trigonometry principles.
Step 1: Draw the isosceles triangle with equal sides of length s, and label the angles and sides as shown below:
/\
/ \
/ \
/__\
s s s
_____
theta
Step 2: Draw the altitude (h) from the top vertex to the base of the triangle, creating two right-angled triangles.
/\
/ \
/ \
/__\
s s s
_____
theta
/ \
/ \
/_____\
h
Step 3: From the right-angled triangle on the left side, we can see that sin(theta) = h/s. Rearranging this equation, we get h = s * sin(theta).
Step 4: The base of the triangle is equal to 2s, as it is formed by the combination of the two equal sides.
Step 5: Now, we can calculate the area (A) of the isosceles triangle using the formula for the area of a triangle: A = (1/2) * base * height.
Substituting the values in, we have:
A = (1/2) * (2s) * (s * sin(theta))
= s^2 * sin(theta)
Therefore, we have proven that the area (A) of an isosceles triangle with equal sides of length s and an angle between them as theta is given by A = (1/2) * s^2 * sin(theta).
To show that the area (A) of an isosceles triangle with equal sides of length s and an angle between them of θ is given by A = (1/2) s^2 sin θ, we can use the formula for the area of a triangle.
The formula for the area of a triangle is A = (1/2) * base * height, where the base is one of the sides, and the height is the perpendicular distance from the base to the opposite vertex.
In an isosceles triangle, the base is one of the equal sides, let's call it s. The height is the perpendicular distance from the base to the opposite vertex.
To find the height, we can use the sine function, as the height can be expressed as h = s * sin θ, where θ is the angle between the equal sides.
Now, substituting the values for the base (s) and height (h) into the formula for the area of a triangle, we have:
A = (1/2) * s * h
= (1/2) * s * (s * sin θ)
= (1/2) * s^2 * sin θ
Therefore, we have shown that the area (A) of an isosceles triangle with equal sides of length s and an angle between them of θ is given by A = (1/2) s^2 sin θ.