we have line QR parallel= to line ST, PQ = 8, QS = , and PT = 12. Find PR.
42
To find PR, we need to understand the given information and the relationship between the lines.
We are told that line QR is parallel to line ST. In a parallel line setup, corresponding angles are congruent, and corresponding sides are proportional.
Given:
PQ = 8
QS = ?
PT = 12
To find PR, we first need to find the value of QS.
Looking at triangle PQS, we have two sides given: PQ = 8 and PT = 12. We need to find QS.
Using the triangle proportionality theorem, we can write the following proportion:
PQ/QS = PT/PS
Substituting the given values into the proportion:
8/QS = 12/PS
Now, we need to manipulate the equation to isolate QS:
Cross-multiplying the equation:
8 * PS = 12 * QS
PS = (12 * QS) / 8
Since QR is parallel to ST, the corresponding sides are proportional. This means that QR/TS = PS/TS.
Substituting the values from the equation:
QR/TS = (12 * QS) / 8
Since QR and TS are parallel, their corresponding sides can be taken as equal:
QR = TS
So, we can rewrite the equation as:
QR/QR = (12 * QS) / 8
Simplifying:
1 = (12 * QS) / 8
Now we can solve for QS:
(12 * QS) / 8 = 1
Cross-multiplying:
12 * QS = 8 * 1
12 * QS = 8
Dividing both sides by 12:
QS = 8/12
QS = 2/3
Now that we have found QS, we can find PR.
To find PR, we use the triangle proportionality theorem again, comparing sides QR and TS.
QR/TS = PS/TS
Since QR and TS are parallel, their corresponding sides are proportional:
QR/TS = PS/TS
Substituting the given values:
QR/TS = PS/TS
Since QR and TS are parallel, their corresponding sides can be taken as equal:
QR = TS
So, we have:
QR/QR = PS/TS
This simplifies to:
1 = PS/TS
Substituting the value of PS:
1 = (12 * QS)/TS
Now, we can solve for PR:
PR = QR + RS
Since QR = TS and RS = QS, we have:
PR = QR + QS
Substituting the known values:
PR = QR + 2/3
Since QR and TS are parallel, their corresponding sides can be taken as equal:
PR = TS + 2/3
Substituting the given value of PT:
PR = 12 + 2/3
Now, we can add the whole numbers and fractions separately:
PR = 12 + 2/3
To add the fractions, we need to have a common denominator. In this case, the common denominator is 3:
PR = 12/1 + (2/3 * 3/3)
PR = 12 * 3/3 + 6/3
PR = 36/3 + 6/3
Adding the numerators:
PR = (36 + 6)/3
PR = 42/3
Finally, we simplify the fraction:
PR = 14
Therefore, PR = 14.