Question

determine the following values. * trapezoid bhln is shown where \\( \overline{bh} \parallel \overline{nl} \\) with midsegment \\( \overline{dp} \\) and \\( bh = 5x + 12 \\), \\( nl = 3x \\), and \\( dp = 5x \\).

Explanation:

Step1: Apply trapezoid midsegment theorem

The midsegment of a trapezoid is equal to half the sum of the two parallel sides. So:
$$DP = \frac{BH + NL}{2}$$

Step2: Substitute given expressions

Replace each variable with its given algebraic form:
$$5x = \frac{(5x + 12) + 3x}{2}$$

Step3: Simplify the right-hand side

Combine like terms in the numerator:
$$5x = \frac{8x + 12}{2}$$
$$5x = 4x + 6$$

Step4: Solve for $x$

Subtract $4x$ from both sides:
$$5x - 4x = 6$$
$$x = 6$$

Step5: Calculate $DP$

Substitute $x=6$ into $DP=5x$:
$$DP = 5 \times 6 = 30$$

Step6: Calculate $BH$

Substitute $x=6$ into $BH=5x+12$:
$$BH = 5 \times 6 + 12 = 30 + 12 = 42$$

Step7: Calculate $NL$

Substitute $x=6$ into $NL=3x$:
$$NL = 3 \times 6 = 18$$

Answer:

$x = 6$
$DP = 30$
$BH = 42$
$NL = 18$