Answer to Question #158206 in Geometry for Janine Leodones

Question

Can two rectangles have:

a.

equal areas and unequal perimeters?

b.

equal perimeters and unequ

al areas?

c.

equal perimeters and unequal diagonals?

d.

equal diagonals, and unequal perimeters?

e.

equal diagonals, equal bases, and unequal perimeters?

f.

equal diagonals, equal altitudes, and unequal areas?

Answer 1

a) The area of ​​the rectangle is found by the formula $S=a*b$ , the perimeter by the formula $P=2(a+b)$, where a and b are the sides of the rectangle. It is obvious that there are pairs of numbers that have an equal product, but a different doubled sum (for example, (6; 6) and (4; 9)), hence, two rectangles can have the equal areas and unequal perimeters.

b) Similar to the previous proof, there are pairs of numbers that have the same doubled sum and different products (for example, (5; 5) and (2; 8)), hence, two rectangles can have the unequal areas and equal perimeters.

c) The diagonal of a rectangle forms a right-angled triangle with its sides, which means that with the same sum of two sides of the rectangles, they can have diagonal lengths that are different

d) In a rectangle, the diagonals are divided by the intersection point in half, hence one side of the rectangle can be found by the formula $a=\sqrt{\frac{d^2}{4}+\frac{d^2}{4}-2*\frac{d}{2}*\frac{d}{2}*cos\alpha }=\sqrt{\frac{d^2}{2}-\frac{d^2}{2}*cos\alpha}$ , where a is the first side of the rectangle, d is the diagonal, $\alpha$ is the angle between the diagonals. And the second side of the rectangle can be found by the formula $b=\sqrt{d^2-a^2}.$

It can be seen from the formulas that the values ​​of the lengths of the sides, and therefore of the perimeter, also depend on the angle between the diagonals, which means that rectangles can have equal diagonals and unequal perimeters.

e) If the rectangles have equal bases, then they also have equal perimeters. Rectnagles cannot have equal diagonals, bases and unequal perimeters.

f) if the rectangles have equal altitudes, then they have equal sides, it follows that they have equal areas. Rectangles cannot have equal diagonals, altitudes and unequal areas