Question

In triangle, N, O, P, comma△NOP, angle, P, \cong, angle, O, comma∠P≅∠O, O, P, equals, 7OP=7 and N, O, equals, 4NO=4. Find P, N, .PN.

Answer 1

In triangle \( \triangle NOP \), we are given that \( \angle P \cong \angle O \), which means that the triangle is isosceles with \( OP = 7 \) and \( NO = 4 \).

Since \( \angle P \cong \angle O \), it follows that the sides opposite these angles are equal. Therefore, we have:

\[
NP = NO
\]

Given \( NO = 4 \), we find that:

\[
NP = 4
\]

Now we also have the side \( OP = 7 \).

At this point, we can summarize the lengths:

- \( OP = 7 \)
- \( NO = 4 \)
- \( NP = 4 \)

To find the length \( PN \), we can use the fact that we have two sides (NO and NP) and the angle opposite the base.

However, this triangle doesn't give us enough information to calculate \( PN \) based purely on the lengths given. We recognize that since \( NP \) and \( NO \) are both equal to 4, and we know \( OP = 7 \), but we have an isosceles triangle with two sides equal to \( NO \).

We can apply the Triangle Inequality Theorem which states that the sum of the lengths of any two sides must be greater than the length of the third side. Thus, with sides \( NP \), \( NO \), and \( OP \):

1. \( NP + NO > OP \) → \( 4 + 4 > 7 \) → True
2. \( NP + OP > NO \) → \( 4 + 7 > 4 \) → True
3. \( NO + OP > NP \) → \( 4 + 7 > 4 \) → True

Since all these inequalities hold, the triangle can exist with the given parameters.

To find the specific lengths, you'd typically apply the Law of Cosines or Sine if needed, but as we currently have:

- \( NP = 4 \)
- \( NO = 4 \)
- \( OP = 7 \)

Thus, based on your requirements, we conclude:

\[
\text{The length of } PN = 4.
\]

Final answer:
\[
PN = 4.
\]