Question

e if $jl = 5x + 1$ and $mk = 8x - 32$, find $jl$ and $nk$.
$jl =$
$nk =$
number:
color:
f if $rv = x + 6$, $us = 5x - 9$, and $rs = 11$, find $vt$ and $st$.
$vt =$
$st =$
number:
color:
@ gina wilson (all things algebra), 2016

Explanation:

Step1: Set diagonals equal (rectangle property)

In rectangle JKLM, diagonals $JL = MK$. So:
$5x + 1 = 8x - 32$

Step2: Solve for x

Rearrange terms:
$1 + 32 = 8x - 5x$
$33 = 3x$
$x = \frac{33}{3} = 11$

Step3: Calculate JL

Substitute $x=11$ into $JL=5x+1$:
$JL = 5(11) + 1 = 55 + 1 = 56$

Step4: Calculate NK (half of JL)

In rectangles, diagonals bisect each other, so $NK = \frac{JL}{2}$:
$NK = \frac{56}{2} = 28$
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Step5: Set sides equal (rectangle property)

In rectangle RSTU, opposite sides $RV = US$? No, wait: diagonals bisect each other, and $RS = UT = 11$. First solve for x using $RV = US$ (diagonals are equal, so their halves are equal):
$x + 6 = 5x - 9$

Step6: Solve for x

Rearrange terms:
$6 + 9 = 5x - x$
$15 = 4x$
$x = \frac{15}{4} = 3.75$

Step7: Calculate VT (equal to RS? No: use Pythagoras? Wait, no: in rectangle, diagonals are equal. First find diagonal length: $RT = 2 \times RV = 2(x+6) = 2(3.75+6) = 2(9.75) = 19.5$

Now use Pythagoras for $ST$: $RT^2 = RS^2 + ST^2$

Step8: Calculate ST

$ST = \sqrt{RT^2 - RS^2} = \sqrt{19.5^2 - 11^2} = \sqrt{380.25 - 121} = \sqrt{259.25} = \frac{\sqrt{1037}}{2} \approx 16.10$

Step9: VT equals RS (opposite sides of rectangle)

$VT = RS = 11$

Answer:

For problem E:
$JL = 56$
$NK = 28$

For problem F:
$VT = 11$
$ST = \frac{\sqrt{1037}}{2} \approx 16.10$