Question

Solve the following problems using two variables.
(v) Seg AB is the diameter of a circle . C is the point on the circumference such that in$∆ABC,\angle B\mathrm{is}\mathrm{less}\mathrm{by}10°\mathrm{than}\angle A$ Find the measure of all the angles of $∆ABC$

Answer 1

$$\begin{gathered} \mathrm{Let}\angle A=x°\mathrm{and}\angle B=y° \\ AB\mathrm{is}\mathrm{diameter}\mathrm{and}C\mathrm{is}\mathrm{a}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{circumference}. \\ \therefore \angle C=90° \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\angle B\mathrm{is}\mathrm{less}\mathrm{than}\angle A\mathrm{by}10°. \\ \mathrm{i}.\mathrm{e}.,\angle B=\angle A-10° \\ \Rightarrow y=x-10 \\ \Rightarrow x-y=10...\left(1\right) \\ \mathrm{Using}\mathrm{the}\mathrm{angle}\mathrm{sum}\mathrm{property},\mathrm{we}\mathrm{have}: \\ \angle A+\angle B+\angle C=180° \\ \Rightarrow x+y+90=180 \\ \Rightarrow x+y=90...\left(2\right) \\ \mathrm{Adding}\mathrm{equation}\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get}: \\ x-y=10 \\ \underline{x+y=90} \\ 2x=100 \\ \therefore x=50 \\ \mathrm{Substituting}x=50\mathrm{in}\mathrm{equation}\left(1\right),\mathrm{we}\mathrm{get}: \\ 50-y=10 \\ \Rightarrow y=40 \\ \therefore \mathrm{The}\mathrm{angles}\mathrm{of}\mathrm{the}\mathrm{triangles}\mathrm{are}40°,50°\mathrm{and}90°. \end{gathered}$$