Question

P, Q, and R are three different points. PQ = 3x + 2, QR = x, RP = x + 2, and . List the angles of PQR in order from largest to smallest and justify your response.

Answer 1

The angles of triangle PQR are ordered from largest to smallest as <>R,

Step-by-step explanation:

The question involves applying the principles of geometry to compare lengths of sides in a triangle, and thereby determine the relative magnitude of angles in triangle PQR. According to the triangle inequality theorem, the largest angle in a triangle is opposite the longest side, and the smallest angle is opposite the shortest side. Given that the side lengths are represented as PQ = 3x + 2, QR = x, and RP = x + 2, we can compare the expressions to conclude which side is longest and which is shortest, assuming all values of x are positive since they represent lengths.

Firstly, it is obvious that QR (x) is the shortest side since it is just x without any additional positive value. Secondly, between RP (x + 2) and PQ (3x + 2), PQ will always be longer than RP for all positive x because it has a larger coefficient in front of x. Hence, the angle opposite PQ (angle R) will be the largest angle, and the angle opposite QR (angle P) will be the smallest. The angle at Q will be between the other two angles in terms of their measurements since PQ is longer than RP but both are longer than QR.

To summarize, the angles of Triangle PQR ordered from largest to smallest are: ∠R, ∠Q, and ∠P.

Answer 2

PQR is a triangle
QR = x ⇒ x>0
If x>0 then:
QR is the smallest side
PQ is the largest side

In the triangle, the largest angle lies opposite the largest side.
The angles of ΔPQR in order from largest to smallest:
∠R is largest [opposite to PQ]
∠Q is middle [opposite to RP]
∠P is smallest [opposite to QR]