In the diagram shown, M, N and P are collinear and QM=QN as shown. If mMQN = 48" and mNQP = 33. Justify why QNP must be isosceles.

An isosceles triangle is that the triangle must have two sides of equal length.

Triangle QNP is isosceles triangle because, QN = PN

In triangle QMN,  

        Since,  QM = QN

 So,  ∠QMN = ∠QNM

By property of triangle:

∠MQN + ∠QNM + ∠QMN = 180

   48 + 2 ∠QNM = 180

              ∠QNM = = 66  degree

  So, ∠QMN = ∠QNM = 66 degree

from figure,

    ∠QNM + ∠QNP = 180

                    ∠QNP = 180 - 66 = 114 degree.

In triangle QNP,  

              ∠QNP + ∠PQN + ∠QPN = 180

                        ∠QPN = 180 - 33 - 114 = 33 degree

Since,     ∠QNP = ∠QPN = 33 degree

Therefore, triangle QNP is isosceles triangle.

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Answer/Step-by-step explanation:

Let's find the measure of the angles of ∆QNP.

∆QMN is am isosceles ∆, because it has two equal sides. Therefore, its base angles would be the same. Thus:

m<MNQ = ½(180 - 48) (one of the base angles of ∆QMN)

m<MNQ = ½(132) = 66°

Next, find m<QNP

m<QNP = 180° - m<MNQ (linear pair angles)

m<QNP = 180° - 66° (Substitution)

m<QNP = 114°

Next, find m<P

m<P = 180 - (m<QNP + m<PQN) (sum of ∆)

m<P = 180 - (114 + 33)

m<P = 180 - 147

m<P = 33°

Thus, in ∆QNP, there are two equal angles, namely, <P and <PQN.

An isosceles ∆ had two equal base angles. Therefore, ∆QNP must be an isosceles ∆.

c. (3,2,6) is the answer.

g = (2, 2)

step-by-step explanation:

given the ratio is 1 : 1 then f is the midpoint of eg

let the coordinates of g = (x, y)

using the midpoint formula

(0 + x) = 1 ( multiply both sides by 2 )

0 + x = 2 ⇒ x = 2

similarly

(4 + y) = 3 ( multiply both sides by 2 )

4 + y = 6 ( subtract 4 from both sides )

y = 2

coordinates of g = (2, 2)