CDE~PQR CD = 9 m EC = 15 what is the length of RP
Since CD = 9 m and EC = 15 m, we can find the length of DE.
DE = EC - CD
DE = 15 m - 9 m
DE = 6 m
Now, we have a right triangle RPQ where PQ is the hypotenuse, PR is the base, and QR is the height.
Using the Pythagorean theorem, we can find the length of PQ.
PQ^2 = PR^2 + QR^2
Since CDE~PQR, we can set up the following proportions:
PQ/DE = PR/CD
PQ/6 = PR/9
Cross-multiplying, we get:
PQ * 9 = 6 * PR
9PQ = 6PR
Dividing both sides by 6, we get:
(9/6)PQ = PR
PQ = (3/2)PR
Substituting this back into the Pythagorean theorem equation, we have:
(3/2)PR^2 = PR^2 + QR^2
Multiplying through by 2, we get:
3PR^2 = 2PR^2 + 2QR^2
Subtracting 2PR^2 from both sides, we get:
PR^2 = 2QR^2
Dividing through by QR^2, we get:
(PR/QR)^2 = 2
Taking the square root of both sides, we get:
PR/QR = sqrt(2)
Since PR/QR = PQ/QR, we can substitute back in the value we found earlier for PQ/QR:
sqrt(2) = (3/2)PR/QR
Since CD = 9 m, and PR/QR = sqrt(2), we can set up the following proportion:
CD/QR = PR/QR
9/QR = sqrt(2)
Cross-multiplying, we get:
9 = QR * sqrt(2)
Dividing both sides by sqrt(2), we get:
QR = 9/sqrt(2)
Multiplying the numerator and denominator of the right-hand side by sqrt(2), we can rationalize the denominator:
QR = (9/sqrt(2)) * (sqrt(2)/sqrt(2))
QR = 9 * sqrt(2) / 2
QR = (9/2) * sqrt(2)
Therefore, the length of QR is (9/2) * sqrt(2) m.