Question

Quadrilateral MNPQ is shown. There is a quadrilateral MNPQ in which side MN is congruent to side PQ and side NP is parallel to side MQ. The diagonal MP and the diagonal NQ intersect each other at R. The length of QR is 4.1. The measure of angle NMP is 30 degrees and the measure of angle QMP is 32 degrees. What is m∠NPQ? A. 62° B. 116° C. 118° D. not enough information

Answer 1

QN = 11 units.

In quadrilateral MNPQ, we're given that side MN is congruent to side PQ, NP is parallel to MQ, and the diagonals MP and NQ intersect at point R.

Given:

MP = 6x - 5

QR = 3x + 1

RN = 6

Since MP is a diagonal, it intersects NQ at point R, so MP = RN + NP (using the property that diagonals of a parallelogram bisect each other).

Given that RN = 6, we can find NP:

MP = RN + NP

6x - 5 = 6 + NP

NP = 6x - 5 - 6

NP = 6x - 11

Now, considering the entire diagonal NQ:

NQ = NP + PQ

NQ = 6x - 11 + PQ

Given MN is congruent to PQ, which implies MN = PQ:

PQ = MN = 6x - 11

Now, let's express QR in terms of PQ:

QR = 3x + 1

We know that QR = QN + RN, and we're given RN = 6:

QR = QN + 6

Let's equate QR in terms of PQ:3x + 1 = QN + 6

Now, substituting PQ = MN = 6x - 11 into the equation:

3x + 1 = QN + 6

3x = QN + 5

Substituting PQ = 6x - 11 into the equation:

3x = 6x - 11 + 5

3x = 6x - 6

3x - 6x = -6

-3x = -6

x = 2

Now that we've found x = 2, let's substitute it back into the expressions to find the lengths:

PQ = MN = 6x - 11

PQ = MN = 6(2) - 11

PQ = MN = 12 - 11

PQ = MN = 1

Now, let's find QN using QN = 3x + 5:

QN = 3x + 5

QN = 3(2) + 5

QN = 6 + 5

QN = 11

Therefore, QN = 11 units.

Question

There is a quadrilateral MNPQ in which side MN is congruent to side PQ and side NP is parallel to side MQ. The diagonal MP and the diagonal NQ intersect each other at point R. If MP = 6x − 5, QR = 3x + 1, and RN = 6, what is QN?