Given: △ABC∼△CDE
BC¯¯¯¯¯¯¯¯
and CD¯¯¯¯¯¯¯¯
are horizontal segments.
AB¯¯¯¯¯¯¯¯
and ED¯¯¯¯¯¯¯¯
are vertical segments.
Lines l
and m
are perpendicular.
Prove: Lines l
and m
have slopes that are opposite reciprocals.
Lines l and m intersect at point C. More information is in the long description.Using the given information, how can it be proved that lines l
and m
have slopes that are opposite reciprocals?
Responses
Since similar triangles have proportional sides, use the proportion ABBC=CDDE.
Show that the slope of l=−DECD
and the slope of m=ABBC.
Then show that CDDE(−DECD)=−1,
so the slopes of l
and m
are opposite reciprocals.
Since similar triangles have proportional sides, use the proportion Show that the slope of l is equal to negative cap d cap e over cap c cap d and the slope of Then show that cap c cap d over cap d cap e times open paren negative cap d cap e over cap c cap d close paren is equal to negative 1 textsf comma so the slopes of l and m are opposite reciprocals.
Since similar triangles have proportional sides, use the proportion BCAB=DECD.
Show that the slope of l=DECD
and the slope of m=−ABBC.
Then show that BCAB(−ABBC)=−1,
so the slopes of l
and m
are opposite reciprocals.
Since similar triangles have proportional sides, use the proportion Show that the slope of l is equal to cap d cap e over cap c cap d and the slope of Then show that cap b cap c over cap A cap b times open paren negative cap A cap b over cap b cap c close paren is equal to negative 1 textsf comma so the slopes of l and m are opposite reciprocals.
Since similar triangles have proportional sides, use the proportion BCAB=DECD.
Show that the slope of l=−CDDE
and the slope of m=BCAB.
Then show that DECD(−CDDE)=−1,
so the slopes of l
and m
are opposite reciprocals.
Since similar triangles have proportional sides, use the proportion Show that the slope of l is equal to negative cap c cap d over cap d cap e and the slope of Then show that cap d cap e over cap c cap d times open paren negative cap c cap d over cap d cap e close paren is equal to negative 1 textsf comma so the slopes of l and m are opposite reciprocals.
Since similar triangles have proportional sides, use the proportion ABBC=CDDE.
Show that the slope of l=CDDE
and the slope of m=−CBBA.
Then show that ABBC(−CBBA)=−1,
so the slopes of l
and m
are opposite reciprocals.
Since similar triangles have proportional sides, we can use the proportion AB/BC = CD/DE.
From this, we can show that the slope of line l is CD/DE and the slope of line m is AB/BC.
Then, by multiplying the two slopes together, (CD/DE)(AB/BC), we get -1 which means the slopes of lines l and m are opposite reciprocals.
This shows that lines l and m have slopes that are opposite reciprocals.