Question

See the attached image and
PROVE PM $\blue{is \: perpendicular \: to}$ XY.

Thanks in Advance !
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Answer 1

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✤ Required Answer:

► GiveN:

• We have to consider a line XY.
• and PM (Perpendicular) smallest line

► To Prove:

• Perpendicular line PM is the shortest line segment.

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✤ How to prove?

By suing triangular inequality, we can prove the above statement that perpendicular is the shortest line segment possible from any point to a line.

• We know that, Shortest angle faces the shortest side and the largest angle faces the largest side in a triangle.

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► Construction:

• Take another point on line XY and let it be N. Now join PN.

► Proof:

Let's take a look on △PMN

➝ ∠M = 90°

We know,

Angle sum property of triangle, All angles add upto 180°

➝ ∠P + ∠M + ∠N = 180°

➝ ∠P + 90° + ∠N = 180°

➝ ∠P + ∠N = 90°

So,

• ∠N < 90°
• ∠N < ∠M

By triangular inequality,

• PM < PN

Similarly, if we take several other points like A, B, C, Q, R,S etc. on the line XY and compare in the same way by using triangle inequality, Then we will confirm that perpendicular will remain the shortest line segment in each case.

• PM < PD
• PM < PQ..... so on$.$

Thus,

Perpendicular is the shortest line segment from a point outside a line to the line.

Hence, proved !!

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Answer 2

Answer:

━━━━━━━━━━━━━━━━━━━━

✤ Required Answer:

► GiveN:

We have to consider a line XY.

and PM (Perpendicular) smallest line

► To Prove:

Perpendicular line PM is the shortest line segment.

━━━━━━━━━━━━━━━━━━━━

✤ How to prove?

By suing triangular inequality, we can prove the above statement that perpendicular is the shortest line segment possible from any point to a line.

We know that, Shortest angle faces the shortest side and the largest angle faces the largest side in a triangle.

━━━━━━━━━━━━━━━━━━━━

► Construction:

Take another point on line XY and let it be N. Now join PN.

► Proof:

Let's take a look on △PMN

➝ ∠M = 90°

We know,

Angle sum property of triangle, All angles add upto 180°

➝ ∠P + ∠M + ∠N = 180°

➝ ∠P + 90° + ∠N = 180°

➝ ∠P + ∠N = 90°

So,

∠N < 90°

∠N < ∠M

By triangular inequality,

PM < PN

Similarly, if we take several other points like A, B, C, Q, R,S etc. on the line XY and compare in the same way by using triangle inequality, Then we will confirm that perpendicular will remain the shortest line segment in each case.

PM < PD

PM < PQ..... so on..

Thus,

Perpendicular is the shortest line segment from a point outside a line to the line.

Hence, proved !!

━━━━━━━━━━━━━━━━━━━━