Question

Triangle P Q R is shown. Angle R P Q is 99 degrees and angle P Q R is 31 degrees. The length of Q R is 11. Determine the measures of all unknown angles and side lengths of ΔPQR. Round side lengths to the nearest hundredth.

Answer 1

Answer:

PR = 5.74 units and

PQ = 8.53 units

Explanation:

We are given the following details:

To find:

We know that the sum of all the angles of a triangle is equal to

i.e.

To find the sides, we can use Sine rule:

As per Sine rule:

Where a, b and c are the sides opposite to respectively.

Using Sine rule in given triangle:

So, sides are

PR = 5.74 units and

PQ = 8.53 units

Explanation:

Answer 2

`\angle PRQ = 50^\circ`

PR = 5.74 units and

PQ = 8.53 units

Explanation:

We are given the following details:

`\angle RPQ =99^\circ`

`\angle PQR =31^\circ\\\text{Side QR} = 11\ units`

To find:

`\angle PRQ =?\\Side\ PR = ?\\Side\ PQ = ?`

We know that the sum of all the angles of a triangle is equal to
`180^\circ`

i.e.

`\angle PQR +\angle PRQ +\angle RPQ =180^\circ\\\Rightarrow 31^\circ +\angle PRQ +99^\circ =180^\circ\\\Rightarrow \angle PRQ = 180-130\\\Rightarrow \angle PRQ = 50^\circ`

To find the sides, we can use Sine rule:

As per Sine rule:

`(a)/(sin A) =(b)/(sin B) =(c)/(sin C)`

Where a, b and c are the sides opposite to
`\angle A,\angle B,\angle C` respectively.

Using Sine rule in given triangle:

`(11)/(sin 99) =(b)/(sin 31) =(c)/(sin 50)\\\\\text{Solving }(11)/(sin 99) =(b)/(sin 31) \\\Rightarrow b = (11)/(sin 99) * sin31\\\Rightarrow b =5.74\ units\\\\\text{Now, Solving }(11)/(sin 99) =(c)/(sin 50) \\\Rightarrow c = (11)/(sin 99) * sin50\\\Rightarrow c =8.53\ units`

So, sides are

PR = 5.74 units and

PQ = 8.53 units