A horseman left the village A at a point with coordinates (2, 5) and began riding along a straight road whose direction was given by the vector v=6i+5j. Then at some point he turned at a right angle; he never changed direction again until he arrived in the village B at the point with coordinates (8, 12). At the point where he made the turn he buried a jar full of silver coins. Unfortunately, he forgot the coordinates of the point. Can you help him recover them?

Accepted Solution

Answer:The point where he  buried his treasure is (9.04,10.74)Step-by-step explanation:First of all, we are going to find the equation  of the line which describes the first path  crossed by the horseman.  In order to find it we have to take in account that the vector V  provides us the direction of this path. We can associate  the direction with the slope of the line. The slope is defined by the ratio of the  vertical changes to horizontal changes between two points.  According to the V=6i+5j  we can determine that:Vertical change (y)= 5Horizontal change (x)=6Slope=\frac{5}{6}Now , using the point- slope form[tex]y-y1=m(x-x1)[/tex]The chosen point is the point where the horseman began riding (2,5). Therefore:m=5/6 y1=5 x1=2y-5=5/6(x-2)y=[tex]\frac{5x}{6}+\frac{10}{3}[/tex]Since the horseman at some point turned at a right angle towards village B  and he  unchanged his direction until arrived in the village B, the second path must be described by a line perpendicular to the first path.  We should know that two lines are perpendicular if and only if their slopes are negative reciprocals This means m1*m2=-1m1=5/6m2=[tex]\frac{-1}{m1} =\frac{-6}{5}[/tex]In order to find the equation of the second path , we will use again the point-slope form.The chosen point is the point where is located the Village B (8,12)[tex]y-12=\frac{-6}{5}(x-8)\\y=\frac{-6x}{5} +21.6[/tex]The coordinates of the point where the horseman buried a jar full of silver coins  corresponds to the intersection of the path 1 and the path 2. Therefore we are going to equal the two equations of each path.[tex]\frac{5x}{6}+\frac{10}{3}=-\frac{6x}{5} +21.6[/tex]Solving this equality for xX=8.98Replacing this value in any of equations of the pathsy=10.74.Finally, the point where he buried his treasure is (9.04,10.74)