Question

if $rv = x + 6$, $us = 5x - 9$, and $rs = 11$, find $vt$ and $st$.
$vt =$
number:
$st =$
color:
© gina wilson (all things algebra), 2016

Explanation:

Step1: Use rectangle diagonal property

In a rectangle, diagonals are equal and bisect each other, so $RV = \frac{1}{2}RT$ and $US = RT$. Thus, $US = 2RV$.
Substitute expressions: $5x - 9 = 2(x + 6)$

Step2: Solve for $x$

Expand and simplify:
$5x - 9 = 2x + 12$
$5x - 2x = 12 + 9$
$3x = 21$
$x = 7$

Step3: Calculate $RV$ then $VT$

Since diagonals bisect each other, $VT = RV$.
$RV = x + 6 = 7 + 6 = 13$, so $VT = 13$.

Step4: Find $ST$ using Pythagoras

$RT = US = 5x - 9 = 5(7) - 9 = 26$.
In right triangle $RST$, $RS=11$, $RT=26$. Use Pythagorean theorem:
$ST = \sqrt{RT^2 - RS^2} = \sqrt{26^2 - 11^2}$
Calculate:
$26^2 = 676$, $11^2 = 121$
$ST = \sqrt{676 - 121} = \sqrt{555}$

Answer:

$VT = 13$
$ST = \sqrt{555}$