Question

if S is a point on side PQ of triangle pqr such that ps=qs=rs

Answer 1

Answer:

∆PQR, by Pythagoras theorem,

PR2 + QR2 = PQ2

Step-by-step explanation:

∆PQR

PS = QS + RS ……(i)

In ∆PSR

PS = RS ….. [from Equation (i)]

⇒ ∠1 = ∠2 Equation ….(ii)

Similarly,

In ∆RSQ,

⇒ ∠3 = ∠4 Equation……(iii)

[Corresponding angles of equal sides are equal]

[By using Equations (ii) and (iii)]

Now in,

∆PQR, sum of angles = 180°

⇒ ∠P + ∠Q + ∠R = 180°

⇒ ∠2 + ∠4 + ∠1 + ∠3 = 180°

⇒ ∠1 + ∠3 + ∠1 + ∠3 = 180°

⇒∠2 (1 + ∠3) = 180°

⇒ ∠1 + ∠3 = (180°)/2 = 90°

∴ ∠R = 90°

Answer 2

Answer:

Given, S is a point on side PQ of a triangle PQR.

Also, PS = QS = RS

If S is a point on side PQ of a △PQR such that PS = QS = RS, then

In triangle PSR,

Given, PS = RS

We know that angles opposite to equal sides in a triangle are equal

∠P = ∠R

So, ∠P = ∠1

Similarly, in triangle RSQ

Given, RS = QS

∠R = ∠Q

So, ∠Q = 2

Considering triangle PQR,

We know that the sum of all three interior angles of a triangle is always equal to 180 degrees.

So, ∠P + ∠Q + ∠PRQ = 180°

∠1 + ∠2 + ∠PRQ = 180°

AAA criterion states that if two angles of a triangle are respectively equal to two angles of another triangle, then by the angle sum property of a triangle their third angle will also be equal.

So, ∠PRQ = ∠1 + ∠2

Now, ∠1 + ∠2 + ∠1 + ∠2 = 180°

2(∠1 + ∠2) = 180°

∠1 + ∠2 = 180°/2

∠1 + ∠2 = 90°

∠PRQ = 90°

So PRQ is a right triangle with right angle at R.

By Pythagoras theorem,

PQ^2 = PR^2 + QR^2