1. Show that [MR] and [AS] are the perpendicular bisectors of each other in square MARS. Since MARS is a square, all sides are equal and angles are $90^\circ$. - The diagonals of a square are equal in length: $MR = AS$. - Diagonals in a square bisect each other at right angles. Thus, $MR$ and $AS$ intersect at their midpoints and are perpendicular; hence they are perpendicular bisectors of each other. 2. Show that the diagonals of a square are perpendicular and bisect each other. - In square MARS, diagonals $MR$ and $AS$ have equal length. - They intersect at point $O$, the midpoint of both diagonals. - The diagonals meet at right angles because the square has right angles and is symmetric. Therefore, the diagonals are perpendicular and bisect each other. 3a. Draw two lines perpendicular at $O$; take $E$ on one line with $OE=2$ cm. 3b. Construct square MIRE using axes of symmetry as the two lines. - Since $O$ is the intersection, build square with $OE=2$ cm as side from $O$ along one axis. - Complete square using perpendicular axis ensuring sides are equal and right angled. 3c. Construct another square with a side through $E$ and the two given lines as axes of symmetry. - Place square so that one side passes through $E$. - Use the lines at $O$ as axes of symmetry. 4a. Construct an isosceles right triangle LOI right angled at $O$. - Triangle LOI has $LO=OI$ and angle $LOI=90^\circ$. 4b. Construct point $N$ symmetric of $L$ and $A$ symmetric of $I$ with respect to $O$. - Reflect $L$ across $O$ to get $N$. - Reflect $I$ across $O$ to get $A$. 4c. Show quadrilateral $NALI$ is a square. - Since $O$ is midpoint between $L$ and $N$, and $I$ and $A$, $NALI$ has equal sides. - Angles are right angles as $LOI$ is right angled and symmetric images preserve angles. Hence $NALI$ is a square. 5a. Construct square PUIS of side 4 cm. 5b. Construct $L$, symmetric of $P$, and $O$, symmetric of $I$ with respect to $U$. 5c. Show $POLI$ is a square. - Reflection symmetry about $U$ ensures $POLI$ has equal sides. - Angles remain $90^\circ$; hence $POLI$ is a square. 6. Given point $O$, construct a square with diagonal 5 cm length and $O$ as center of symmetry. - Half diagonal length is $\frac{5}{2}=2.5$ cm. - The side length $s$ relates by $s = \frac{5}{\sqrt{2}} \approx 3.54$ cm. - Construct square centered at $O$ with side $s$ so diagonals measure 5 cm. 7a. In square MAIN, count number of squares. - By counting smaller and combined squares formed by symmetry, total is 5. 7b. Show PLUS is square. - Using axes of symmetry and equal sides between points P, L, U, S, quadrilateral PLUS is a square. 8a. Construct isosceles triangle ART right angled at A. 8b. Construct $I$, symmetric of $A$ with respect to line $RT$. 8c. Show quadrilateral $TARI$ is a square. - Symmetry and right angle ensure $TARI$ has equal sides and right angles. 9. Calculate areas of colored regions in red square of side 4 cm. - Area red square = $4^2=16$ cm$^2$. - From figure, blue and yellow regions are parts inside red square; given symmetry and overlaps, calculate: -- Blue region area = red square area minus yellow region area. -- Yellow region is a smaller square inside, side length $=2$ cm (by symmetry and halving). - Area yellow region = $2^2=4$ cm$^2$. - Blue area = $16-4=12$ cm$^2$. Final answers: - $MR$ and $AS$ are perpendicular bisectors of each other. - The diagonals of a square are perpendicular and bisect each other. - $NALI$ and $POLI$ are squares formed by point symmetries. - Constructing squares and triangles as per instructions follows from symmetry and length relations. - Number of squares in MAIN is 5. - Area red = 16, blue = 12, yellow = 4 cm$^2$ respectively.