Question

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 1

The figure with the triangle ABC and the correct statement, along with the answer choices, is attached.

As per it, the answer is: The length of DC is 28

Step-by-step explanation:

From the figure and the information that segment AB = segment AC, you know that the base angles of the isosceles triangle ABC are B and C and they are congruent.

The small triangles BED and CFD have two pairs of congruent angles (the angle B is congruent to angle D, and the angle BED is congruent to angle DFC), so triangles BED and CFD are similar.

The similarity of those two triangles leads to the following proportional relation

• DE / DF = DB / DC

And the ratio DE to DF is given as 5 : 7; thus:

• 5/7 = DB / DC

It is also given that segment BC = 48.

From the figure:

• BC = DB + DC = 48.

Hence, you have these two equations:

• 5/7 = DB / DC

• DB + DC = 48.

Where the two unknowns are DB and BC.

To facilitate operations, make DC = x, which means that DB = 48 - x, and the resultant equation is:

• 5 /7 = (48 - x) / x

Solve:

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

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

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

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

• x = 28 . . . [simplification]

Therefore, length of segment x = DC = 28.