Math Is Fun Forum

Olivia cut a rectangle into exactly nine squares. On inspection, he observed that the area of one square was 64 cm²
, the areas of two other squares were 16 cm²
, and the rest of them were 4 cm²
each.
What was the perimeter of the original rectangl

The opposite sides of a convex hexagon

are parallel, the distances separating the parallel pairs of sides are equal,
and the angles at vertices A and D are right angles. Prove that the diagonals BE
and CF enclose an angle of

Find, without calculus, the largest possible value of

(sin 5x + cos 3x)/(sin 4x + cos 4x)

Prove that if two right-angled triangles

have the same perimeter and the same area,
then they are congruent.

find x

In an acute-angled triangle ABC, a tangent is drawn to the inscribed circle, parallel to side BC. The tangent intersects side AC at point D. F is the orthogonal projection of point D onto the side BC. Show that AB=AD+BF

hard

from Mr Παύλος

M,N,Z,I midpoint

find the area of  EMBN

What is the ordered pair of real numbers (x,y) for which
123x+ 321y= 345
and

321x+ 123y= 543?