Question

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Answer 1

The arrangement of the sides in order from shortest to longest is PR , QR , PQ .

Explanation:

Consider the given triangle PQR, with ∠P = (12n-9)° , ∠Q = (62-3n)°

and ∠R = (16n+2)°.

We have to arrange the sides in order from shortest to longest.

Side opposite to the largest angle is the longest.

But to apply this, we first need to find the measure of each angle,

We know, Angle sum property of a triangle states that "the sum of angles of a triangle is equal to 180°".

∠P + ∠Q + ∠R = 180°

Substitute the values, we get,

(12n-9)° + (62-3n)° + (16n+2)° = 180°

Combining same terms together, we get,

⇒ 12n -3n+16n + 62 -9 +2 = 180

⇒ 25n + 55 = 180

⇒ 25n = 180 - 55

⇒ 25 n = 125

⇒ n = 5

Thus, ∠P = (12n-9)° = (12(5)-9) = 51°

∠Q = (62-3n)° = (62-3(5))° = 47°

and ∠R = (16n+2)° = (16(5)+2)° = 82°

Thus, ∠R is the greatest angle. thus side opposite to ∠R is longest that is PQ is longest.

Then ∠P is the middle angle , so side opposite to ∠P is Middle that is QR is middle.

Then ∠Q is the smallest angle , so side opposite to ∠Q is smallest that is PR is smallest.

Thus, The arrangement of the sides in order from shortest to longest is PR , QR , PQ .

Answer 2

The correct option is b.

Explanation:

It is given that

`\angle P=(12n-9)^(\circ)`

`\angle Q=(62-3n)^(\circ)`

`\angle R=(16n+2)^(\circ)`

According to the angle sum property, the sum of interior angles of a triangle is 180 degree.

`\angle P+\angle Q+\angle R=180^(\circ)`

`12n-9+62-3n+16n+2=180^(\circ)`

`25n+55=180^(\circ)`

`25n=125^(\circ)`

`n=5`

Th value of n is 5, so the measure of angles are

`\angle P=(12(5)-9)=51^(\circ)`

`\angle Q=(62-3(5))=47^(\circ)`

`\angle R=(16(5)+2)=82^(\circ)`

`R>P>Q`

In a triangle, the largest angle has longest opposite side and smallest angle has shortest opposite side

`PQ>QR>PR`

The order from shortest to longest is PR, QR, PQ. Option b is correct.