$$\sin 45°=\frac{h_c}{b}$$

$$h_c=b·\sin 45°$$

$$h_c=14·\frac{\sqrt{2}}{2}=7·\bcancel{2}·\frac{\sqrt{2}}{\bcancel{2}}$$

$$a^2=b^2+c^2-2bc·\cos 45°$$

$${10}^2={14}^2+c^2-\bcancel{2}·14·c·\frac{\sqrt{2}}{\bcancel{2}}$$

$$100=196+c^2-14\sqrt{2}c$$

$$196+c^2-14\sqrt{2}c-100=0$$

$$c^2-14\sqrt{2}c+96=0$$

$$c_{1,2}=\frac{-\left(-14\sqrt{2}\right)\pm \sqrt{{\left(14\sqrt{2}\right)}^2-4·1·96}}{2}$$

$$c_{1,2}=\frac{14\sqrt{2}\pm \sqrt{196·2-384}}{2}$$

$$c_{1,2}=\frac{14\sqrt{2}\pm \sqrt{8}}{2}$$

$$c_{1,2}=\frac{14\sqrt{2}\pm \sqrt{2·4}}{2}=\frac{14\sqrt{2}\pm \sqrt{2}\sqrt{4}}{2}$$

$$c_{1,2}=\frac{14\sqrt{2}\pm 2\sqrt{2}}{2}$$

$$c_1=\frac{14\sqrt{2}+2\sqrt{2}}{2}=\frac{16\sqrt{2}}{2}=\frac{8·\bcancel{2}\sqrt{2}}{\bcancel{2}}$$

$$c_2=\frac{14\sqrt{2}-2\sqrt{2}}{2}=\frac{12\sqrt{2}}{2}=\frac{6·\bcancel{2}\sqrt{2}}{\bcancel{2}}$$

$$A_1=\frac{c_1·h_c}{2}=\frac{8\bcancel{\sqrt{2}}·7\bcancel{\sqrt{2}}}{\bcancel{2}}=8·7$$

$$A_2=\frac{c_2·h_c}{2}=\frac{6\bcancel{\sqrt{2}}·7\bcancel{\sqrt{2}}}{\bcancel{2}}=6·7$$