Question

e triangles below are similar. find the value of x. the image is not to scale.
two triangles: triangle efd with sides ef=30, fd=35, ed=60; triangle bca with sides bc=x, ca=63, ba=108, angles at f and c are equal, angles at e and b are equal, angles at d and a are equal

Explanation:

Step1: Set up proportion for similar triangles

Since the triangles are similar, the ratios of corresponding sides are equal. So, $\frac{EF}{BC}=\frac{FD}{CA}=\frac{ED}{BA}$. We can use $\frac{EF}{BC}=\frac{ED}{BA}$ (or other corresponding sides). Let's take $\frac{30}{x}=\frac{60}{108}$? Wait, no, wait. Wait, let's check the corresponding sides. Wait, triangle EFD and triangle BCA? Wait, angle at F and angle at C are equal (the marked angles), and angle at E and angle at B, angle at D and angle at A. So sides: EF corresponds to BC, FD corresponds to CA, ED corresponds to BA. So EF = 30, FD = 35, ED = 60; BC = x, CA = 63, BA = 108. So the ratio of EF to BC should equal FD to CA and ED to BA. Let's use FD/CA = ED/BA? Wait, FD is 35, CA is 63; ED is 60, BA is 108. Let's check 35/63 = 5/9, 60/108 = 5/9. So that's correct. Now, EF is 30, BC is x. So 30/x should also be 5/9? Wait, no, wait: EF corresponds to BC, FD to CA, ED to BA. So EF/BC = FD/CA. So 30/x = 35/63. Let's simplify 35/63 = 5/9. So 30/x = 5/9. Cross-multiplying: 5x = 309 = 270. Then x = 270/5 = 54? Wait, wait, maybe I mixed up the correspondence. Wait, let's re-examine. The first triangle: E---F---D, with EF=30, FD=35, ED=60. The second triangle: B---C---A, with BC=x, CA=63, BA=108. Angles at F and C are equal (the two arcs), angles at E and B, D and A. So the order is E-F-D and B-C-A. So side EF (between E and F) corresponds to BC (between B and C), side FD (between F and D) corresponds to CA (between C and A), side ED (between E and D) corresponds to BA (between B and A). So the ratios: EF/BC = FD/CA = ED/BA. So EF=30, BC=x; FD=35, CA=63; ED=60, BA=108. Let's check FD/CA: 35/63 = 5/9. ED/BA: 60/108 = 5/9. So that's consistent. Now EF/BC: 30/x should equal 5/9. So 30/x = 5/9. Cross multiply: 5x = 309 = 270. Then x = 270/5 = 54. Wait, but let's check with another ratio. FD/CA = 35/63 = 5/9, EF/BC = 30/54 = 5/9. Yes, that works. Alternatively, ED/BA = 60/108 = 5/9. So that's correct.

Step2: Solve for x

From the proportion $\frac{30}{x} = \frac{35}{63}$, simplify $\frac{35}{63}$ to $\frac{5}{9}$. So we have $\frac{30}{x} = \frac{5}{9}$. Cross - multiply: $5x = 30\times9$. Calculate $30\times9 = 270$. Then $x=\frac{270}{5}=54$.

Answer:

54