We know, Sum of all interior angles of a triangle is 180°.
So, Using this property of triangles, we have
[tex] \sf \:\angle O + \angle R + \angle P = {180}^{ \circ} \\ [/tex]
[tex] \sf \:\angle O + \angle R + {90}^{ \circ} = {180}^{ \circ} \\ [/tex]
[tex] \sf \:\angle O + \angle R = {180}^{ \circ} - {90}^{ \circ} \\ [/tex]
[tex]\implies\sf\:\angle O + \angle R = {90}^{ \circ} \\ [/tex]
Thus, from these calculations, we concluded that
[tex]\implies\sf\:\angle P > \angle R \\ [/tex]
We know, Side opposite to greater angle is longest.
[tex]\implies\sf\:OR > OP \\ [/tex]
[tex]\implies\bf\:OP < OR \\ [/tex]
Thus, the perpendicular drawn from an exterior point is the shortest distance from the point to the base line.
Similarly for all the other triangles, in the similar manner as above, we concluded that the perpendicular drawn from an exterior point is the shortest distance from the point to the base line.
Hence, the perpendicular drawn from an exterior point is the shortest distance from the point to the base line.
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[tex]\large\underline{\sf{Solution-}}[/tex]
In right-angle triangle ORP
We know, Sum of all interior angles of a triangle is 180°.
So, Using this property of triangles, we have
[tex] \sf \:\angle O + \angle R + \angle P = {180}^{ \circ} \\ [/tex]
[tex] \sf \:\angle O + \angle R + {90}^{ \circ} = {180}^{ \circ} \\ [/tex]
[tex] \sf \:\angle O + \angle R = {180}^{ \circ} - {90}^{ \circ} \\ [/tex]
[tex]\implies\sf\:\angle O + \angle R = {90}^{ \circ} \\ [/tex]
Thus, from these calculations, we concluded that
[tex]\implies\sf\:\angle P > \angle R \\ [/tex]
We know, Side opposite to greater angle is longest.
[tex]\implies\sf\:OR > OP \\ [/tex]
[tex]\implies\bf\:OP < OR \\ [/tex]
Thus, the perpendicular drawn from an exterior point is the shortest distance from the point to the base line.
Similarly for all the other triangles, in the similar manner as above, we concluded that the perpendicular drawn from an exterior point is the shortest distance from the point to the base line.
Hence, the perpendicular drawn from an exterior point is the shortest distance from the point to the base line.