Final answer:
To find the lengths of sides QR and PR in similar triangles ABC and PQR, we can use the property that corresponding sides of similar triangles are in proportion to each other. Substituting the given side lengths, we can set up two equations and solve for QR and PR. The lengths of sides QR and PR are QR = (8/15) * (2/3) * PQ and PR = (15/8) * QR, respectively.
Step-by-step explanation:
To find the lengths of sides QR and PR, we can use the property that corresponding sides of similar triangles are in proportion to each other. In triangle ABC, the sides are AB = 24, BC = 40, and AC = 32. In triangle PQR, the corresponding sides will have the same ratios. So, using the ratios, we can find the lengths of QR and PR.
AB/QR = BC/PR = AC/PQ
Substituting the given values, we have:
24/QR = 40/PR = 32/PQ
From this, we can set up two equations:
24/QR = 40/PR
40/PR = 32/PQ
Cross multiplying these equations, we get:
24PR = 40QR
40PQ = 32PR
Simplifying these equations gives:
PR = (24/40) * QR
PQ = (40/32) * PR
Substituting the PR value in the second equation, we get:
PQ = (40/32) * (24/40) * QR
QR = (32/40) * (40/24) * PQ
Simplifying the equations gives:
PR = (3/5) * QR
PQ = (10/8) * PR
Substituting the PR value in the second equation, we get:
PQ = (10/8) * (3/5) * QR
QR = (5/8) * (8/3) * PQ
Simplifying further gives:
PR = (15/8) * QR
PQ = (3/2) * PR
Substituting the PR value in the second equation, we get:
PQ = (3/2) * (15/8) * QR
QR = (8/15) * (2/3) * PQ
So, the lengths of sides QR and PR are QR = (8/15) * (2/3) * PQ and PR = (15/8) * QR, respectively.