using thepythagorean theorem and if the quadrilateral with side c is a square.then compute thearea of the square with side a + b:(a) as (a + b)2 and (b) as the sum of the areas of the 4 triangles and the square withside c.
To find the area of a square with side length (a + b) using the Pythagorean theorem, we can use the formula (a + b)^2.
(a) Computing area using (a + b)^2:
1. Start with the formula for the area of a square: A = s^2, where A represents the area and s represents the side length of the square.
2. Substitute (a + b) for s in the formula: A = (a + b)^2.
3. Expand the square: A = (a + b) * (a + b).
Applying the distributive property, we get: A = a^2 + ab + ba + b^2.
4. Simplify the expression: A = a^2 + 2ab + b^2.
The area of the square with side length (a + b) is given by the simplified expression a^2 + 2ab + b^2.
(b) Computing area using triangles and a square:
1. Consider a square with side length c. The area of this square is given by A = c^2.
2. Divide the square into four triangles by drawing two diagonal lines from opposite corners.
3. Each triangle will have a base of length a or b and a height of length c.
4. The area of each triangle can be calculated using the formula for the area of a triangle: A_triangle = 0.5 * base * height.
So, A_triangle = 0.5 * a * c for each of the two triangles with a as the base.
Similarly, A_triangle = 0.5 * b * c for each of the two triangles with b as the base.
5. Now, sum up the area of the square and the four triangles to get the total area:
A_total = A_square + 2 * A_triangle + 2 * A_triangle.
= c^2 + 2 * 0.5 * a * c + 2 * 0.5 * b * c
= c^2 + ac + bc.
So, the area of the square with side length (a + b) can be computed as (a) (a + b)^2 and (b) c^2 + ac + bc.