# Two Squares +-- {.image} [[TwoSquares.png:pic]] > The red square has double the area of the yellow square. What's the angle? =-- ## Solution by [[square|Properties of Squares]] and [[Angle at the Circumference is Half the Angle at the Centre]] +-- {.image} [[TwoSquaresLabelled.png:pic]] =-- As the red square has double the area of the yellow square, the [[square|length of the diagonal]] of the yellow square is the same as the side length of the red square. This means that a circle drawn with centre $C$ through $A$ also passes through $B$ and $D$. Then as the [[angle at the circumference is half the angle at the centre]], angle $A \hat{D} B$ is half of $A \hat{C} B$, which is $90^\circ$. So angle $A \hat{D} B$ is $45^\circ$. ## Solution by [[square|Properties of Squares]], [[Isosceles Triangle]], and [[Angles in a Triangle]] As above $C D$ has the same length as $C B$ so triangle $C B D$ is [[isosceles]]. Since angle $B \hat{C} D$ is $45^\circ$, this means that angle $C \hat{D} B$ is $67.5^\circ$. Triangle $A D C$ is also [[isosceles]] and angle $A \hat{C} D$ is $135^\circ$ so angle $C \hat{D} A$ is $22.5^\circ$. This makes angle $A \hat{D} B$ equal to $67.5^\circ - 22.5^\circ = 45^\circ$.