Question
Triangle P R Q is shown. The length of side P R is 5 n. The length of side R Q is 32 + n. Angles R P Q and P Q R are congruent.
What is the length of PR?
units
What is the length of PR?
units
Answers
GPT-4o mini
To find the length of side PR in triangle PQR given the information about the sides and the congruent angles, we can use the property that when two angles in a triangle are congruent, the sides opposite those angles are also equal.
Since angles RQ and RPQ are congruent, the sides opposite those angles (which are PR and QR) must be equal. Therefore, we can set up the following equation based on the lengths of the sides:
\[ PR = QR \]
Given:
- \( PR = 5n \)
- \( QR = 32 + n \)
Setting them equal gives us:
\[ 5n = 32 + n \]
To solve for \( n \), we can first subtract \( n \) from both sides:
\[ 5n - n = 32 \]
This simplifies to:
\[ 4n = 32 \]
Next, we divide both sides by 4:
\[ n = 8 \]
Now that we have the value of \( n \), we can substitute it back into the expression for the length of side PR:
\[ PR = 5n = 5 \times 8 = 40 \]
Thus, the length of side PR is:
\[ \boxed{40} \text{ units} \]
Since angles RQ and RPQ are congruent, the sides opposite those angles (which are PR and QR) must be equal. Therefore, we can set up the following equation based on the lengths of the sides:
\[ PR = QR \]
Given:
- \( PR = 5n \)
- \( QR = 32 + n \)
Setting them equal gives us:
\[ 5n = 32 + n \]
To solve for \( n \), we can first subtract \( n \) from both sides:
\[ 5n - n = 32 \]
This simplifies to:
\[ 4n = 32 \]
Next, we divide both sides by 4:
\[ n = 8 \]
Now that we have the value of \( n \), we can substitute it back into the expression for the length of side PR:
\[ PR = 5n = 5 \times 8 = 40 \]
Thus, the length of side PR is:
\[ \boxed{40} \text{ units} \]