Question

determine the following values.
isosceles trapezoid rvdx is shown with midsegment \\( \overline{bm} \\) where \\( rv = 2x + 1 \\), \\( xd = 3x + 2 \\), \\( bm = 2x + 4 \\) and \\( md = 3 \\).
(the image shows the isosceles trapezoid rvdx with points r, v, d, x and midsegment bm, and a table with numbers 3,5,7,8,9,11,12,14,15 and boxes for x=, bm=, xd=, rv=, and the perimeter of rvdz)

Explanation:

Step1: Use midsegment property

In an isosceles trapezoid, the midsegment length is the average of the two base lengths, and also, since $B$ and $M$ are midpoints, $XD = RV + 2\times MD$. Substitute given expressions:
$$3x + 2 = (2x + 1) + 2\times3$$

Step2: Simplify and solve for x

Simplify the right-hand side, then isolate $x$:
$$3x + 2 = 2x + 1 + 6$$
$$3x + 2 = 2x + 7$$
$$3x - 2x = 7 - 2$$
$$x = 5$$

Step3: Calculate BM

Substitute $x=5$ into $BM=2x+4$:
$$BM = 2(5) + 4 = 10 + 4 = 14$$

Step4: Calculate XD

Substitute $x=5$ into $XD=3x+2$:
$$XD = 3(5) + 2 = 15 + 2 = 17$$

Step5: Calculate RV

Substitute $x=5$ into $RV=2x+1$:
$$RV = 2(5) + 1 = 10 + 1 = 11$$

Step6: Calculate perimeter of RVDX

The legs of the isosceles trapezoid are equal, so $RX = VD$. Since $B$ and $M$ are midpoints, $RX = 2\times MB_{segment}$? No, use the fact that $MD=3$, so the leg length is $2\times MD = 6$ (because $M$ is the midpoint of $VD$, $B$ midpoint of $RX$). Perimeter is sum of all sides:
$$Perimeter = RV + VD + DX + XR = 11 + 6 + 17 + 6 = 40$$

Answer:

$x=5$
$BM=14$
$XD=17$
$RV=11$
Perimeter of RVDX = 40