Exercise 38 Page 579

Let's analyze the given figure.

Since we are given a triangle with three lines that are parallel to one of its sides, we can use the Triangle Proportionality Theorem and its Corollary. The lengths of the segments intercepted by the parallel line are proportional. Let's write these proportions. ML/LK=MP/PQ, LK/KJ=PQ/QR, KJ/JH=QR/RS By substituting the already known lengths into the first proportion, we can find ML. ML/LK=MP/PQ ⇕ ML/4=3/6We will solve it for ML.

To find QR, we have to simplify the second proportion. LK/KJ=PQ/QR ⇕ 4/2=6/QR Let's solve it.

Since we know QR, we can find JH from the third proportion. KJ/JH=QR/RS ⇕ 2/JH=3/6 Let's do it.

Since QK is parallel to PL, according to the Corresponding Angles Theorem ∠ MLP is congruent to ∠ MKQ, and ∠ MPL is congruent to ∠ MQK. Therefore, by the Angle-Angle Similarity Theorem, △ QKM is similar to △ PLM. Let's write a proportion using the lengths of the sides of both triangles. QK/PL=MK/ML Since we know the lengths of ML and LK, we can find the length of MK.

Now we can substitute all known lengths into our proportion. QK/PL=MK/ML ⇕ QK/2=6/2 Let's solve it to find QK.

Finally, we found that ML=2, QR=3, QK=6, and JH=4.