The convex pentagon ABCDE has a right angle at B,AB=BC andCD=DE=EA=1cm,AC=2cm.If AC and ED are parallel ,find the area of the pentagon.
To find the area of the pentagon, we can divide it into a rectangle and a triangle.
First, let's find the length of AB and BC. Since AB = BC, and we know that AC = 2cm, we can use the Pythagorean theorem to find AB or BC:
AC^2 = AB^2 + BC^2
2^2 = AB^2 + AB^2 (since AB = BC)
4 = 2AB^2
AB^2 = 2
AB = √2 cm
Now that we have the length of AB, we can find the area of the rectangle. The width of the rectangle is AB (√2 cm), and the length is AC (2 cm).
Area of the rectangle = width × length = √2 cm × 2 cm = 2√2 cm^2
Next, let's find the area of the triangle. The base of the triangle is DE (1 cm), and the height is AB (√2 cm).
Area of the triangle = (base × height) / 2 = (1 cm × √2 cm) / 2 = (√2 / 2) cm^2
Finally, we can add the area of the rectangle and the triangle to find the total area of the pentagon:
Total area = Area of rectangle + Area of triangle
Total area = 2√2 cm^2 + (√2 / 2) cm^2
Total area = (4√2 + √2) / 2 cm^2
Total area = (5√2 / 2) cm^2
Therefore, the area of the pentagon is (5√2 / 2) cm^2.
To find the area of the convex pentagon ABCDE, we can split it into two triangles: ABC and CDE.
First, let's find the height of triangle ABC. Since it is a right-angled triangle, we can use the Pythagorean theorem. Let h be the height of triangle ABC.
From the given information, AB = BC = 1 cm and AC = 2 cm. Therefore, using the Pythagorean theorem, we have:
AC^2 = AB^2 + BC^2
2^2 = 1^2 + h^2
4 = 1 + h^2
h^2 = 3
h = √3
Now, let's find the area of triangle ABC. The area of a triangle is given by the formula:
Area = 1/2 * base * height
In this case, the base of triangle ABC is AB = 1 cm
So, the area of triangle ABC is:
Area_ABC = 1/2 * AB * h
= 1/2 * 1 * √3
= √3/2 cm^2
Next, let's find the area of triangle CDE. The base of triangle CDE is DE = 1 cm, and the height is given by h.
The area of triangle CDE is:
Area_CDE = 1/2 * DE * h
= 1/2 * 1 * √3
= √3/2 cm^2
Now, to find the area of the pentagon ABCDE, we add the areas of triangles ABC and CDE:
Area_pentagon = Area_ABC + Area_CDE
= √3/2 + √3/2
= (2√3)/2
= √3 cm^2
Therefore, the area of the convex pentagon ABCDE is √3 cm^2.