If point [tex]P[/tex] is [tex]\frac{4}{7}[/tex] of the distance from [tex]M[/tex] to [tex]N[/tex], what ratio does the point [tex]P[/tex] partition the directed line segment from [tex]M[/tex] to [tex]N[/tex] into?
A. [tex]4: 1[/tex]
B. [tex]4: 3[/tex]
C. [tex]4: 7[/tex]
D. [tex]4: 10[/tex]
Answer (2)
Point P divides the line segment MN such that MP = 7 4 MN .
Express PN as MN − MP , which gives PN = 7 3 MN .
Find the ratio PN MP = 7 3 MN 7 4 MN .
Simplify the ratio to get the final answer: 4 : 3 .
Explanation
Problem Analysis Let M and N be two points on a line. Point P is located on the directed line segment MN such that the distance from M to P is 7 4 of the total distance from M to N . We want to find the ratio in which P divides the directed line segment MN .
Define distances Let MP denote the distance from M to P , and PN denote the distance from P to N . We are given that MP = 7 4 MN . We need to find the ratio MP : PN .
Express PN in terms of MN and MP Since MP + PN = MN , we can express the length of PN in terms of MN .
PN = MN − MP
Substitute MP Substitute MP = 7 4 MN into the equation for PN :
PN = MN − 7 4 MN = 7 7 MN − 7 4 MN = 7 3 MN
Find the ratio MP:PN Now we find the ratio MP : PN by dividing MP by PN :
PN MP = 7 3 MN 7 4 MN
Simplify the ratio Simplify the ratio by canceling out the common factor 7 1 MN :
PN MP = 3 4
Final Answer Express the ratio in the form a : b , which is 4 : 3 . Therefore, the point P partitions the directed line segment MN in the ratio 4 : 3 .
Examples
In architecture, when designing a bridge or a road, engineers often need to divide a line segment into specific ratios to determine the placement of support structures or expansion joints. For instance, if a bridge deck is divided in a 4:3 ratio, it means that one section is 4/7 of the total length, while the other is 3/7. This ensures balanced load distribution and structural integrity.
Point P divides the directed line segment from M to N in the ratio of 4:3. This is found by calculating the distances from M to P and from P to N, given that the distance from M to P is 7 4 of the total distance. Therefore, the answer is option B.
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