Final answer:
In a rhombus, the square of the side times 4 equals the sum of the squares of the diagonals due to the Pythagorean theorem
Step-by-step explanation:
In mathematics, a rhombus, or diamond, is a quadrilateral whose four sides all have the same length. To prove that in a rhombus ABCD, 4ab square = ac square + bd square, we need to recall the Pythagorean theorem and the properties of a rhombus.
Let us assume the measures of half the diagonals are 'p' and 'q' whereas 'a' is the side of the rhombus. As such, the diagonal lengths for the rhombus are 2p and 2q.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the other two sides. In a rhombus, the diagonals intersect at right angles, creating four right triangles within the rhombus.
Applying the Pythagorean theorem to one of these triangles, we get a2=(p2+q2) . If we multiply both sides of the equation by 4, we get 4a2=4p2+4q2. We know that 4p2 is the square of the first diagonal (AC) and 4q2 is the square of the second diagonal (BD). So, we can conclude that 4ab square = ac square + bd square.
Learn more about Properties of Rhombus