# Consider rectangel CREB that is formed of squares COAB and OREA, where AE=4ft. Assume squares COAB and OREA are congruent. BO=3m-2nAR=7(m-1)-4.8nDetermine the values of m and n and the length of RB.

Question
Analytic geometry

Consider rectangel CREB that is formed of squares COAB and OREA, where $$AE=4ft$$. Assume squares COAB and OREA are congruent.
$$BO=3m-2n$$
$$AR=7(m-1)-4.8n$$
Determine the values of m and n and the length of RB.

2021-01-05

The diagonal of each square divides it into two triangles with angles of $$\displaystyle{45}^{\circ},{45}^{\circ}{\quad\text{and}\quad}{90}^{\circ}$$. The diagonals (BO and AR) are the triangles' hypotenuses. The hypotenuses is $$\displaystyle\sqrt{{2}}$$ times the leg in such triangles. A leg is also a square side, which is 4ft, so
$$\displaystyle{B}{O}={A}{R}={4}\sqrt{{2}}$$
$$\displaystyle{3}{m}-{2}{n}={7}{\left({m}-{1}\right)}-{4.8}{n}={4}\sqrt{{2}}$$
$$\displaystyle{3}{m}-{2}{n}={4}\sqrt{{2}}$$
$$\displaystyle-{2}{n}=-{3}{n}+{4}\sqrt{{2}}$$
$$\displaystyle{n}={1.5}{m}-{2}\sqrt{{2}}$$
$$\displaystyle{7}{\left({m}-{1}\right)}-{4.8}{n}={4}\sqrt{{2}}$$
$$\displaystyle{7}{\left({m}-{1}\right)}-{4.8}{\left({1.5}{m}-{2}\sqrt{{2}}\right)}={4}\sqrt{{2}}$$
$$\displaystyle{7}{m}-{7}-{7.2}{m}+{9.6}\sqrt{{2}}={4}\sqrt{{2}}$$
$$\displaystyle-{0.2}{m}={4}\sqrt{{2}}+{7}-{9.6}\sqrt{{2}}$$
$$\displaystyle-{0.2}{m}=-{5.6}\sqrt{{2}}+{7}$$
$$\displaystyle{m}={28}\sqrt{{2}}-{35}$$
$$\displaystyle{n}={1.5}{m}-{2}\sqrt{{2}}$$
$$\displaystyle={1.5}{\left({28}\sqrt{{2}}-{35}\right)}-{2}\sqrt{{2}}$$
$$\displaystyle={42}\sqrt{{2}}-{52.5}-{2}\sqrt{{2}}$$
$$\displaystyle={40}\sqrt{{2}}-{52.5}$$
$$\displaystyle{R}{B}^{{2}}={B}{A}^{{2}}+{A}{E}^{{2}}$$
$$\displaystyle={8}^{{2}}+{4}^{{2}}$$
$$=64+16$$
$$=80$$
$$\displaystyle{R}{B}=\sqrt{{80}}$$
$$\displaystyle={4}\sqrt{{5}}$$

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