Use the x-intercept method to find all real solutions of the equation.
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It has been proven that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.
<h3>How to prove a Line Segment?</h3>We know that in a triangle if one angle is 90 degrees, then the other angles have to be acute.
Let us take a line l and from point P as shown in the attached file, that is, not on line l, draw two line segments PN and PM. Let PN be perpendicular to line l and PM is drawn at some other angle.
In ΔPNM, ∠N = 90°
∠P + ∠N + ∠M = 180° (Angle sum property of a triangle)
∠P + ∠M = 90°
Clearly, ∠M is an acute angle.
Thus; ∠M < ∠N
PN < PM (The side opposite to the smaller angle is smaller)
Similarly, by drawing different line segments from P to l, it can be proved that PN is smaller in comparison to all of them. Therefore, it can be observed that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.
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The coordinates of point P along AB so that the ratio of AP to PB is 3 to 1 is (4,13).
<h3>What are coordinates?</h3>A coordinate system in geometry is a system that employs one or more integers, or coordinates, to define the position of points or other geometric components on a manifold such as Euclidean space.
Given that the ratio of the two lines is 3:1, therefore, we can write,
m:n = 3:1
Also, the coordinate of the endpoints of the line are,
A = (x₁, y₁) = (1, 7)
B = (x₂, y₂) = (5, 15)
Now, Using the Section formula the coordinate of the point P can be written as,
Hence, the coordinates of point P along AB so that the ratio of AP to PB is 3 to 1 is (4,13).
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