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The ratio in which the point (1, 3) divides the line segment joining the points ( – 1, 7) and (4, – 3) is
- a)2 : 3
- b)2 : 7
- c)3 : 2
- d)7: 2
Correct answer is option 'A'. Can you explain this answer?
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Ratio of Dividing a Line Segment
To find the ratio in which a point divides a line segment, we can use the section formula. The section formula states that if a point (x, y) divides the line segment joining two points (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates of the point can be found using the following formulas:
x = (mx2 + nx1) / (m + n)
y = (my2 + ny1) / (m + n)
Given Information:
Point A (1, 7)
Point B (4, 3)
Point P (1, 3)
We need to find the ratio in which point P divides the line segment AB.
Step 1: Calculate the distance between points A and B.
Distance formula:
d = √((x2 - x1)² + (y2 - y1)²)
d = √((4 - 1)² + (3 - 7)²)
= √(3² + (-4)²)
= √(9 + 16)
= √25
= 5
Step 2: Calculate the distance between points A and P.
d1 = √((x - x1)² + (y - y1)²)
d1 = √((1 - 1)² + (3 - 7)²)
= √(0 + (-4)²)
= √(0 + 16)
= √16
= 4
Step 3: Calculate the distance between points B and P.
d2 = √((x - x2)² + (y - y2)²)
d2 = √((1 - 4)² + (3 - 3)²)
= √((-3)² + 0)
= √(9 + 0)
= √9
= 3
Step 4: Calculate the ratio m:n.
m = d1/d = 4/5
n = d2/d = 3/5
The ratio in which point P divides the line segment AB is 4:5.
Step 5: Convert the ratio to its simplest form.
Divide both the terms of the ratio by their greatest common divisor.
4:5 ÷ 1 = 4:5
Therefore, the ratio in which the point (1, 3) divides the line segment joining the points (1, 7) and (4, 3) is 4:5, which corresponds to option A.
To find the ratio in which a point divides a line segment, we can use the section formula. The section formula states that if a point (x, y) divides the line segment joining two points (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates of the point can be found using the following formulas:
x = (mx2 + nx1) / (m + n)
y = (my2 + ny1) / (m + n)
Given Information:
Point A (1, 7)
Point B (4, 3)
Point P (1, 3)
We need to find the ratio in which point P divides the line segment AB.
Step 1: Calculate the distance between points A and B.
Distance formula:
d = √((x2 - x1)² + (y2 - y1)²)
d = √((4 - 1)² + (3 - 7)²)
= √(3² + (-4)²)
= √(9 + 16)
= √25
= 5
Step 2: Calculate the distance between points A and P.
d1 = √((x - x1)² + (y - y1)²)
d1 = √((1 - 1)² + (3 - 7)²)
= √(0 + (-4)²)
= √(0 + 16)
= √16
= 4
Step 3: Calculate the distance between points B and P.
d2 = √((x - x2)² + (y - y2)²)
d2 = √((1 - 4)² + (3 - 3)²)
= √((-3)² + 0)
= √(9 + 0)
= √9
= 3
Step 4: Calculate the ratio m:n.
m = d1/d = 4/5
n = d2/d = 3/5
The ratio in which point P divides the line segment AB is 4:5.
Step 5: Convert the ratio to its simplest form.
Divide both the terms of the ratio by their greatest common divisor.
4:5 ÷ 1 = 4:5
Therefore, the ratio in which the point (1, 3) divides the line segment joining the points (1, 7) and (4, 3) is 4:5, which corresponds to option A.
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The ratio in which the point (1, 3) divides the line segment joining the points ( – 1, 7) and (4, – 3) isa)2 : 3b)2 : 7c)3 : 2d)7: 2Correct answer is option 'A'. Can you explain this answer?
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