Name: Illustration Based on the SAS Criterion
Uploaded: 2019-07-19
Duration: 4 min 36 s
Description: One of the easiest ways to learn a concept is through a visual illustration. To learn how the SAS congruence criterion is applied in mathematical problems, w...

Question

Of all the line segments that can be drawn to a given line, from a point, not lying on it, prove that the perpendicular line segment is the shortest.

Accepted Answer

Let $l$ be any line and $A$ is a point not on the line and $B, C$ are the points on the line from point $A$ . So, $A B ⊥ line l$ .

So, $∠ A B C = 90 °$ -------(i)

Now, in $∆ A B C$ , $∠ A B C + ∠ B A C + ∠ B C A = 180 °$

$\Rightarrow 90 ° + ∠ B A C + ∠ B C A = 180 °$

$\Rightarrow ∠ B A C + ∠ B C A = 90 °$ -----(ii)

So, from equation (i) and (Ii),

$∠ A B C > ∠ B C A$

We know that, in a triangle, the side opposite to the larger angle is longer.

So, $A C > A B$ .

So, of all the line segments that can be drawn to a given line, from a point, not lying on it, the perpendicular line segment is the shortest.

Hence, proved.

Of all the line segments that can be drawn to a given line, from a point, not lying on it, prove that the perpendicular line segment is the shortest.

Important Questions on Triangles

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