What are the values of x and y?
∆ABD : AB = 17, AD = 15
∆BCD : BD = 8 , BC = y , DC = x
A. x = 136/15, y = 17/15
B. x = 64/15, y = 17/15
C. x = 8/15, y = 136/15
D. x = 64/15, y = 136/15
Let's use the Law of Cosines on ∆ABD:
cos(∠ADB) = (AB² + AD² - BD²) / 2(AB)(AD)
cos(∠ADB) = (17² + 15² - 8²) / 2(17)(15)
cos(∠ADB) = (289 + 225 - 64) / 510
cos(∠ADB) = 450 / 510
Now let's use the Law of Cosines on ∆BCD:
cos(∠BDC) = (BC² + DC² - BD²) / 2(BC)(DC)
cos(∠BDC) = (y² + x² - 8²) / 2(y)(x)
Notice that the angle ∠ADB and ∠BDC form a straight angle (∠ADC). Thus, we have:
cos(∠BDC) = -cos(∠ADB)
Now we can set up an equation:
-450 / 510 = (y² + x² - 64) / 2(y)(x)
Solve for one of the variables in terms of the other. Let's solve for y:
y² + x² - 64 = -900(y)(x) / 510
510(y² + x² - 64) = -900(y)(x)
Let's solve for x in terms of y:
x = sqrt(64 - y² + 900(y)/510)
The side DC (x) must be positive, so we need to find the value of y that will make the square root have a positive value. We can try different values of y by using the answer choices:
A. y = 17/15
x = sqrt(64 - (17/15)² + 900(17/15)/510)
x = sqrt(64 - 2.2667 + 2.2667)
x = sqrt(63.9998)
x ≈ 8
So we have x ≈ 8 and y ≈ 17/15, which corresponds to answer choice A:
x = 136/15 and y = 17/15.