Question

If the point P(2,1) lies on the line segment joining the points A(4,2) and B(8,4) then prove that AP = 1/2 AB

Answer 1

Statement:

If the point P(2,1) lies on the line segment joining the points A(4,2) and B(8,4), prove that $\sf{\overline{AP}=\dfrac{1}{2} \overline{AB}}$.

Let's start the proof:

To prove, we need the lengths of $\sf{\overline{AP}}$ and $\sf{\overline{AB}}$.

To find their lengths of $\sf{\overline{AP}}$ and $\sf{\overline{AB}}$, we can apply Pythagorean Theorem, or namely distance formula.

• $\sf{\overline{AB}^2=4^2+2^2}$
• $\sf{\overline{AP}^2=2^2+1^2}$

So their lengths will be the below ones.

• $\sf{\overline{AB}=\sqrt{4^2+2^2} =2\sqrt{5} }$
• $\sf{\overline{AP}=\sqrt{2^2+1^2}=\sqrt{5}}$

Since the equation $\sf{\sqrt{5} =\dfrac{1}{2}\times 2\sqrt{5} }$ is true,

now we have proven that $\sf{\overline{AP}=\dfrac{1}{2} \overline{AB}}$.

For more:

The distance formula finds the distance between the two coordinates.

To start generalization, we set two points A, B to be the below ones.

• $\sf{A(x_1,y_1)}$
• $\sf{B(x_2,y_2)}$

Now for the start of the proof, there are total two cases:

• (1) $\sf{x_1=x_2\:or\:y_1=y_2}$ [or: satisfying at least one.]
• (2) $\sf{x_1\neq x_2\:and\:y_1\neq y_2}$ [and: satisfying all both.]

First case.

Firstly, we solve case (1).

After solving the length of two lines we get the below ones.

• $\sf{\overline{AP}=|x_1-x_2|}$
• $\sf{\overline{BP}=|y_1-y_2|}$

Whether the lengths are 0 or not, Pythagorean Theorem is satisfied, hence the formula too.

Though any triangles form when the length is 0. The Pythagorean Theorem is satisfied whenever.

Second case.

We solve case (2). It will be the following.

• $\sf{\overline{AB}^2=\overline{AP}^2+\overline{BP}^2}$

→ $\sf{\overline{AB}^2=|x_1-x_2|^2+|y_1-y_2|^2}$

→ $\sf{\overline{AB}^2=(x_1-x_2)^2+(y_1-y_2)^2}$ [Properties of Reals]

→ $\sf{\therefore{\overline{AB}=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}}}$ [Positive Distance]

Therefore the distance is:

$\boxed{\sf{\overline{AB}=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2} }}$ .

Answer 2

Step-by-step explanation:

AB

2

=4

2

+2

2

$\sf{\overline{AP}^2=2^2+1^2}$

AP

2

=2

2

+1

2