Math  /  Geometry

QuestionGiven a trapezoid QRST where M and P are the midpoints of the legs, and PM=2x,QR=3x,TS=10, find PM.\text{Given a trapezoid } QRST \text{ where } M \text{ and } P \text{ are the midpoints of the legs, and } PM = 2x, \, QR = 3x, \, TS = 10, \text{ find } PM.

Studdy Solution

STEP 1

1. QRST QRST is a trapezoid with QR QR and TS TS as the parallel sides (bases).
2. M M and P P are midpoints of the non-parallel sides (legs) of the trapezoid.
3. PM PM is the segment connecting the midpoints of the legs.
4. PM=2x PM = 2x , QR=3x QR = 3x , and TS=10 TS = 10 .

STEP 2

1. Recall the Midsegment Theorem for trapezoids.
2. Set up the equation using the Midsegment Theorem.
3. Solve for x x .
4. Calculate PM PM .

STEP 3

Recall the Midsegment Theorem for trapezoids:
The midsegment of a trapezoid is parallel to the bases and its length is the average of the lengths of the two bases.
PM=QR+TS2 PM = \frac{QR + TS}{2}

STEP 4

Set up the equation using the Midsegment Theorem:
2x=3x+102 2x = \frac{3x + 10}{2}

STEP 5

Solve for x x :
Multiply both sides by 2 to eliminate the fraction:
4x=3x+10 4x = 3x + 10
Subtract 3x 3x from both sides:
x=10 x = 10

STEP 6

Calculate PM PM :
Since PM=2x PM = 2x , substitute x=10 x = 10 :
PM=2×10=20 PM = 2 \times 10 = 20
The length of PM PM is:
20 \boxed{20}

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