Question

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1. if $overline{rt}$ bisects $overline{su}$, find each measure.

$st = _____$
$ru = _____$
$sv = _____$
$su = _____$

1. find $x$
2. find $x$
3. find $mn$
4. find $cd$
5. find $jl$
6. if $overline{qt}$ is the perpendicular bisector of $overline{pr}$, find each measure.

$x = _____$
$y = _____$
$pq = _____$
$qr = _____$
$ps = _____$
$sr = _____$
$pt = _____$
$pr = _____

Explanation:

Problem 1

Step1: Find $ST$ (congruent segment)

Since $\overline{RT}$ bisects $\overline{SU}$ at right angles, $ST = RT = 23$

Step2: Find $RU$ (congruent segment)

Segments from bisector to endpoints: $RU = RS = 8$

Step3: Find $SV$ (given segment)

$SV$ is given as $5$

Step4: Find $SU$ (double $SV$)

$SU = 2 \times SV = 2 \times 5 = 10$

Problem 2

Step1: Set segments equal (angle bisector)

Perpendiculars from bisector are equal: $24x - 19 = 16x + 13$

Step2: Solve for $x$

$24x - 16x = 13 + 19$
$8x = 32$
$x = \frac{32}{8} = 4$

Problem 3

Step1: Set segments equal (angle bisector)

Perpendiculars from bisector are equal: $8x - 37 = 5x + 17$

Step2: Solve for $x$

$8x - 5x = 17 + 37$
$3x = 54$
$x = \frac{54}{3} = 18$

Problem 4

Step1: Set sides equal (isosceles triangle)

$\overline{MP}=\overline{PO}$, so $4x + 19 = 6x - 7$

Step2: Solve for $x$

$19 + 7 = 6x - 4x$
$26 = 2x$
$x = 13$

Step3: Calculate $MN$

$MN = 4x + 19 = 4(13) + 19 = 52 + 19 = 71$

Problem 5

Step1: Set segments equal (perpendicular bisector)

$\overline{AD}=\overline{CD}$, so $9x - 15 = 7x - 1$

Step2: Solve for $x$

$9x - 7x = -1 + 15$
$2x = 14$
$x = 7$

Step3: Calculate $CD$

$CD = 7x - 1 = 7(7) - 1 = 49 - 1 = 48$

Problem 6

Step1: Set segments equal (angle bisector)

Segments from bisector to sides are equal: $3x + 4 = 5x - 16$

Step2: Solve for $x$

$4 + 16 = 5x - 3x$
$20 = 2x$
$x = 10$

Step3: Calculate $JL$

$JL = 3x + 4 = 3(10) + 4 = 30 + 4 = 34$

Problem 7

Step1: Solve for $x$ (congruent segments)

$\overline{PS}=\overline{SR}$, so $4x + 4 = 7x - 17$
$4 + 17 = 7x - 4x$
$21 = 3x$
$x = 7$

Step2: Solve for $y$ (congruent segments)

$\overline{PQ}=\overline{QR}$, so $5y - 31 = 2y + 5$
$5y - 2y = 5 + 31$
$3y = 36$
$y = 12$

Step3: Calculate $PQ$

$PQ = 5y - 31 = 5(12) - 31 = 60 - 31 = 29$

Step4: Calculate $QR$

$QR = PQ = 29$

Step5: Calculate $PS$

$PS = 4x + 4 = 4(7) + 4 = 28 + 4 = 32$

Step6: Calculate $SR$

$SR = PS = 32$

Step7: Calculate $PT$

$PT = 6x - 2y = 6(7) - 2(12) = 42 - 24 = 18$

Step8: Calculate $PR$

$PR = 2 \times PT = 2 \times 18 = 36$

Answer:

1. $ST=23$, $RU=8$, $SV=5$, $SU=10$
2. $x=4$
3. $x=18$
4. $MN=71$
5. $CD=48$
6. $JL=34$
7. $x=7$, $y=12$, $PQ=29$, $QR=29$, $PS=32$, $SR=32$, $PT=18$, $PR=36$