Question

1 points s and t are midpoints. if pq = 6, pr = 8, and st = 35, find the length of qr.

1. d, e, and f are midpoints. if ab = 11, bc = 16 and ac = 14, find the perimeter of adef.
2. aabc is equilateral and points d, e, and f are midpoints. if ac = 9, find the perimeter of parallelogram defc.
3. de is a midsegment. if de = 4x - 2 and bc find the value of x.

Explanation:

Problem 1:

Step1: Use the mid - segment theorem

The mid - segment theorem states that \(ST=\frac{1}{2}QR\).

Step2: Solve for \(QR\)

Given \(ST = 35\), then \(QR=2\times ST\).
Since \(ST = 35\), we have \(QR=2\times35\).

Step1: Use the mid - segment theorem

By the mid - segment theorem:

• \(DE=\frac{1}{2}BC\), \(EF=\frac{1}{2}AB\), \(DF=\frac{1}{2}AC\)

Step2: Calculate the lengths of \(DE\), \(EF\), and \(DF\)

Given \(AB = 11\), \(BC=16\), \(AC = 14\)
\(DE=\frac{1}{2}\times16 = 8\), \(EF=\frac{1}{2}\times11=5.5\), \(DF=\frac{1}{2}\times14 = 7\)

Step3: Calculate the perimeter of \(\triangle DEF\)

The perimeter \(P=DE + EF+DF\)
\(P=8 + 5.5+7\)

Step1: Use the mid - segment theorem and properties of an equilateral triangle

Since \(\triangle ABC\) is equilateral with \(AC = 9\), then \(AB=BC = AC=9\)
By the mid - segment theorem: \(DE=\frac{1}{2}BC\), \(EF=\frac{1}{2}AB\), \(DF=\frac{1}{2}AC\), \(FC=\frac{1}{2}AC\)
In parallelogram \(DEFC\), \(DE = FC\) and \(DF=EC\)
\(DE=\frac{1}{2}\times9 = 4.5\), \(DF=\frac{1}{2}\times9=4.5\)

Step2: Calculate the perimeter of parallelogram \(DEFC\)

The perimeter \(P = 2(DE + DF)\)
\(P=2(4.5+4.5)\)

Answer:

\(QR = 70\)

Problem 2: