Bunuel wrote:

In ∆ PQR above, VW is parallel to PR, and S, T, U divide PR into four equal parts. What is the ratio of (length of JL)/(length of VW)?

(A) 1/3

(B) 1/2

(C) 2/3

(D) 3/4

(E) cannot be determined

Attachment:

2017-08-15_1312_001.png

kumarparitosh123, I agree with you. Short version:

All four small triangles (e.g. VQJ) inside triangle QVW are similar to all four of their larger counterpart triangles (e.g. PQS) inside triangle PQR.

Parallel lines cut by transversals create two pairs of congruent corresponding angles; and for each set of triangles, the third angle at Q is shared.

PR is divided into four equal segments. To maintain same ratio of corresponding sides of similar triangles, VW also must be divided into four equal segments. Each segment is \(\frac{1}{4}\) of total length of bases PR and VW.

Length of JL = \(\frac{2}{4}\) = \(\frac{1}{2}\)

Length of PR = \(\frac{4}{4}\) = 1.

Ratio of the two is \(\frac{1}{2}\) : 1 = \(\frac{1}{2}\)

Answer B

If that explanation was cryptic (and if it wasn't, stop reading) . . .

Segmented base of PR creates two similar triangles, one small and one large, where corresponding sides must be in same ratio.

Parallel lines PR and VW are cut by five transversal segments QP, QS, QT, QU, and QR.

Every small triangle (e.g. triangle VQJ) is similar to its larger counterpart (e.g. PQS) because the two triangles have three congruent angles.

Triangles VQJ and PQS have three pairs of congruent angles:

1) they share angle PQS;

2) parallel segments cut by transversal segments QP and QS create corresponding angles that are congruent --

angles QVJ and QPS are congruent corresponding angles, as are angles QJV and QSP.

AAA = similar triangles, which means corresponding sides will have the same ratio.

PR is divided into four equal parts. Let each segment of the base of large triangle PQR be 2x. PR looks like

P__\(_{2x}\)__S__\(_{2x}\)__T__\(_{2x}\)__U__\(_{2x}\)__R

For ratios of corresponding sides to be equal (e.g. PS to JV, ST to JK, etc.), each segment of VW must be equal. Let each segment = x.

V___\(_x\)____J__\(_x\)___K___\(_x\)___L___\(_x\)___W

The ratio of (length of JL)/(length of VW):

\(\frac{JL}{VW}\) = \(\frac{2x}{4x}\) = \(\frac{1}{2}\)

Answer B

*And - the ratio of (length of JL)/(length of VW) is identical to the ratio of (length of SU)/(length of PR).

SU = 4x

PR = 8x

\(\frac{SU}{PR}\) = \(\frac{4x}{8x}\) = \(\frac{1}{2}\)

_________________

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