Same angle
angle, E, F, D, \cong, angle, B, C, D
∠EFD≅
∠BCD
start overline, E, F, end overline, \parallel, start overline, B, C, end overline
EF
∥
BC
: corresponding angles
triangle, E, D, F, \sim, triangle, B, D, C
△EDF∼
△BDC
AA:Two pairs of angles are congruent
start fraction, E, D, divided by, B, D, end fraction, equals, start fraction, E, F, divided by, B, C, end fraction
BD
ED
=
BC
EF
Corresponding sides of similar figures are proportionate.
start fraction, 16, divided by, x, plus, 16, end fraction, equals, start fraction, x, plus, 2, divided by, 15, end fraction
x+16
16
=
15
x+2
Substitute the values
left bracket, x, plus, 16, right bracket, left bracket, x, plus, 2, right bracket, equals, left bracket, 16, right bracket, left bracket, 15, right bracket
(x+16)(x+2)=
(16)(15)
Cross Multiply
x, squared, plus, 2, x, plus, 16, x, plus, 32, equals, 240
x
2
+2x+16x+32=
240
FOIL
x, squared, plus, 18, x, plus, 32, equals, 240
x
2
+18x+32=
240
Combining like terms
x, squared, plus, 18, x, minus, 208, equals, 0
x
2
+18x−208=
0
Subtract 240 from both sides
left bracket, x, minus, 8, right bracket, left bracket, x, plus, 26, right bracket, equals, 0
(x−8)(x+26)=
0
Factoring
x, minus, 8, equals, 0, o, r, x, plus, 26, equals, 0
x−8=0or
x+26=0
Zero Product Rule
x, equals, 8, o, r, x, equals, minus, 26
x=8or
x=−26
x, equals, 8
x=
8
Ignore negative root, as length must be positive.
The length of , start overline, B, E, end overline, is , 8
The length of
BE
is 8
The angle ∠EFD is congruent to ∠BCD and EF is parallel to BC. The triangles ΔEDF and ΔBDC are similar by the AA postulate. By the proportionality of corresponding sides, we can set up the equation BD/ED = BC/EF.
Substituting the values given, we have (16/x + 16) = (x + 2)/15.
Cross-multiplying, we get (x + 16)(x + 2) = (16)(15).
Expanding and simplifying, we have x^2 + 18x + 32 = 240.
Combining like terms, we have x^2 + 18x - 208 = 0.
Subtracting 240 from both sides, we have x^2 + 18x - 208 - 240 = 0.
Simplifying further, we get x^2 + 18x - 448 = 0.
Factoring, we have (x - 8)(x + 26) = 0.
Setting each factor equal to zero, we have x - 8 = 0 or x + 26 = 0.
Solving, we get x = 8 or x = -26.
Since length cannot be negative, we ignore the negative root and conclude that the length of BE is 8.