Question

triangle stu is formed by connecting the midpoints of the side of triangle pqr. the lengths of the sides of triangle stu are shown. find the perimeter of triangle pqr. figures not necessarily drawn to scale.

Explanation:

Step1: Identify Midsegment Theorem

The Midsegment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. Here, \( S, T, U \) are midpoints, so \( ST, TU, US \) are midsegments of \( \triangle PQR \).

Step2: Find Side Lengths of \( \triangle PQR \)

- For side \( PQ \): Corresponding midsegment is \( TU = 3 \), so \( PQ = 2 \times TU = 2 \times 3 = 6 \).
- For side \( QR \): Corresponding midsegment is \( US = 3 \), so \( QR = 2 \times US = 2 \times 3 = 6 \). Wait, no, wait—wait, looking at the diagram, \( ST = 2 \), \( TU = 3 \), \( US = 3 \)? Wait, no, let's re - examine. Wait, \( S, T, U \) are midpoints of \( PQR \), so:
- If \( ST \) is a midsegment, then the side it's parallel to (say \( PR \)) has length \( 2\times ST = 2\times2 = 4 \)? Wait, no, maybe I misassigned. Wait, the lengths of \( \triangle STU \) are: \( ST = 2 \), \( TU = 3 \), \( US = 3 \)? Wait, no, the problem says "the lengths of the sides of triangle \( STU \) are shown". Let's assume \( ST = 2 \), \( TU = 3 \), \( SU = 3 \). Then, since \( S, T, U \) are midpoints, each side of \( \triangle PQR \) is twice the length of the corresponding side of \( \triangle STU \).

Wait, correction: The Midsegment Theorem (also called the Midline Theorem) states that in a triangle, the segment connecting the midpoints of two sides is parallel to the third side and half its length. So, if \( S, T, U \) are midpoints of \( PQ, QR, RP \) (or any combination), then:

- Side \( PQ \): If \( TU \) is the midsegment parallel to \( PQ \), then \( PQ = 2\times TU \).
- Side \( QR \): If \( US \) is the midsegment parallel to \( QR \), then \( QR = 2\times US \).
- Side \( PR \): If \( ST \) is the midsegment parallel to \( PR \), then \( PR = 2\times ST \).

From the diagram, let's take the lengths of \( \triangle STU \): \( ST = 2 \), \( TU = 3 \), \( SU = 3 \).

So:
- \( PR = 2\times ST = 2\times2 = 4 \)
- \( PQ = 2\times TU = 2\times3 = 6 \)
- \( QR = 2\times SU = 2\times3 = 6 \)

Wait, no, that can't be. Wait, maybe the sides of \( \triangle STU \) are \( ST = 2 \), \( TU = 3 \), \( US = 3 \). Then the sides of \( \triangle PQR \) are:

- Corresponding to \( ST \): length \( 2\times ST = 4 \)
- Corresponding to \( TU \): length \( 2\times TU = 6 \)
- Corresponding to \( US \): length \( 2\times US = 6 \)

Now, perimeter of \( \triangle PQR \) is \( PQ + QR + PR \). Wait, maybe I mixed up the labels. Let's re - label: Let \( U \) be midpoint of \( PR \), \( T \) midpoint of \( QR \), \( S \) midpoint of \( PQ \). Then:

- \( ST \) is midsegment of \( \triangle PQR \), parallel to \( PR \), so \( PR = 2\times ST \). Given \( ST = 2 \), so \( PR = 4 \).
- \( TU \) is midsegment of \( \triangle PQR \), parallel to \( PQ \), so \( PQ = 2\times TU \). Given \( TU = 3 \), so \( PQ = 6 \).
- \( SU \) is midsegment of \( \triangle PQR \), parallel to \( QR \), so \( QR = 2\times SU \). Given \( SU = 3 \), so \( QR = 6 \).

Step3: Calculate Perimeter of \( \triangle PQR \)

Perimeter \( = PQ + QR + PR = 6 + 6 + 4 = 16 \)? Wait, no, wait, maybe I made a mistake. Wait, the sides of \( \triangle STU \): if \( ST = 2 \), \( TU = 3 \), \( US = 3 \), then:

Wait, another approach: The perimeter of \( \triangle STU \) is \( 2 + 3 + 3 = 8 \). Since \( \triangle STU \) is the medial triangle of \( \triangle PQR \), the perimeter of \( \triangle PQR \) is twice the perimeter of \( \triangle STU \). Wait, is that a property? Yes! The medial triangle (formed by midpoints) has a perimeter half of the original triangle. Wait, no: Wait, the medial triangle's sides are half the length of the original triangle's sides. So if the original triangle has sides \( a, b, c \), the medial triangle has sides \( \frac{a}{2},\frac{b}{2},\frac{c}{2} \). So perimeter of medial triangle is \( \frac{a + b + c}{2} \), so perimeter of original triangle is \( 2\times \) perimeter of medial triangle.

Perimeter of \( \triangle STU \): \( 2 + 3 + 3 = 8 \). Then perimeter of \( \triangle PQR = 2\times8 = 16 \)? Wait, no, wait, if \( ST = 2 \), \( TU = 3 \), \( US = 3 \), then perimeter of \( STU \) is \( 2 + 3 + 3 = 8 \), so perimeter of \( PQR \) is \( 2\times8 = 16 \)? Wait, but let's check with midsegment:

If \( ST \) is midsegment, length \( 2 \), so the side it's parallel to is \( 4 \). \( TU = 3 \), so the side it's parallel to is \( 6 \). \( US = 3 \), so the side it's parallel to is \( 6 \). Then perimeter is \( 4 + 6 + 6 = 16 \). Yes, that matches.

Wait, but maybe I misread the lengths. Let's assume the lengths of \( \triangle STU \) are \( ST = 2 \), \( TU = 3 \), \( SU = 3 \). Then:

- Side parallel to \( ST \): \( 2\times2 = 4 \)
- Side parallel to \( TU \): \( 2\times3 = 6 \)
- Side parallel to \( SU \): \( 2\times3 = 6 \)

Perimeter \( = 4 + 6 + 6 = 16 \).

Wait, but let's re - check the problem statement: "Triangle \( STU \) is formed by connecting the midpoints of the side of triangle \( PQR \). The lengths of the sides of triangle \( STU \) are shown. Find the perimeter of triangle \( PQR \)."

So if \( ST = 2 \), \( TU = 3 \), \( US = 3 \), then:

Perimeter of \( \triangle STU = 2 + 3 + 3 = 8 \)

Since \( \triangle STU \) is the medial triangle, perimeter of \( \triangle PQR = 2\times \) perimeter of \( \triangle STU = 2\times8 = 16 \).

Answer:

The perimeter of triangle \( PQR \) is \( \boldsymbol{16} \).