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
please answer 💀
To solve this problem, we can use proportions of corresponding sides from similar triangles. Since triangles ABD and BCD share angle B, we can say the triangles are similar (∆ABD ~ ∆BCD) if angle A is congruent to angle BCD.
Using the Side-Splitter Theorem, which states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally:
AD/DC = AB/BC
Plugging in the given values:
15/x = 17/y
Next, we know the side lengths of triangle ABD, so we can use the Pythagorean theorem to find the length of BD:
AB^2 + AD^2 = BD^2
17^2 + 15^2 = 289 + 225 = 514
BD^2 = 514
BD = √514
Now we can use the proportion:
AD/DC = BD/BC
15/x = √514/y
We also have:
BD = 8
√514 = 8
Now we can set up a system of equations:
15/x = 17/y eq. 1
√514/y = 8 eq. 2
From eq. 2, we can find the value of y:
y = √514/8
y = 136/15
Now, plugging the value of y into eq. 1:
15/x = 17/(136/15)
Solve for x:
x = 15 * 17 * 15 / 136
x = 64/15
Answer: (x, y) = (64/15, 136/15), so the correct answer is D. x = 64/15, y = 136/15.