Triangle abc above is isosceles with an=ac and bc=48. The ratio of de to de is 5:7 . What is the length of dc

Answer:

The diagram displaying triangle ABC and the relevant statement along with the answer options is attached.

According to this, the answer is: The length of DC is 28.

Explanation:

From the diagram and the data that segment AB equals segment AC, you comprehend that angles B and C of the isosceles triangle ABC are congruent.

Triangles BED and CFD possess two sets of congruent angles (angle B is congruent to angle D, and angles BED and DFC are congruent), indicating that triangles BED and CFD are similar.

This similarity leads to the proportional relationship:

• DE / DF = DB / DC

Given that the ratio of DE to DF is 5:7, we have:

• 5/7 = DB / DC

Also, it is stated that segment BC is 48.

From the figure:

• BC = DB + DC = 48.

This results in two equations:

• 5/7 = DB / DC

• DB + DC = 48.

Now, we define DC = x, which implies DB = 48 - x, leading to the equation:

• 5 / 7 = (48 - x) / x

Solve this:

• 5x = 7 (48 - x)... [multiplication property: multiply by x and 7]

• 5x = 336 - 7x... [distributive property]

• 12x = 336... [addition property: add 7x to both sides]

• x = 336 / 12... [division property: divide by 12]

• x = 28... [simplification]

Therefore, the length of segment x = DC = 28.