Question

When deriving the distance formula, Eowyn starts with two points, A(x1,y1) and B(x2,y2). She then draws point C on a horizontal line from point A and a vertical line from point B, giving it coordinates (x2,y1). In doing so, she believes she created a right triangle, with right angle C, which allows her to use the Pythagorean Theorem. She lets AC=b and BC=a, which means that AB=c by definition of the hypotenuse. Next, she uses the Ruler Postulate to state that a=|x2−x1| and b=|y2−y1|. Then, combining this information with the Pythagorean Theorem, she is able to use the Substitution Property of Equality to state that (x2−x1)2+(y2−y1)2=AB2. Finally, she applies the Square Root Property of Equality to find AB=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√. She is able to ignore the possible negative root since distance is always positive, by definition. Which statement about Eowyn's derivation is the most accurate?

A )Eowyn correctly derived the distance formula. She properly supported each statement and used theorems, properties, and postulates to correctly justify her reasoning.

B) Eowyn didn't correctly derive the distance formula. She should have drawn point C so that it had coordinates C(x1,y2), which changes her values of a and b, as well as the set up of AB2.

C) Eowyn didn't correctly derive the distance formula. She drew in her auxiliary segments correctly, but she didn't define a, b, and c correctly in order to apply them in the Pythagorean Theorem.

D) Eowyn didn't correctly derive the distance formula. She drew in her auxiliary segments correctly, and properly defined her variables, but she misused the Ruler Postulate to find values of a and b in terms of x and y.

Answer 1

Explanation:

the answer is a because it is thanks i hope you get it right

Answer 2

Answer with explanation:

The two Starting Points used by Eowyn for deriving Distance formula:

`=A(x_(1),y_(1)) \text{and} B(x_(2),y_(2))`

Then Eowyn has Drawn the point C

→ She Draw a Horizontal Line From Point A, and a vertical line from point B, and then the point where these lines intersect was named the point
`C(x_(3),y_(3))`.

→ She Supposed, AC=b, BC=a

and then, AB=c

→Ruler Postulate

`a=|x_(2)-x_(1)|\\\\b=|y_(2)-y_(1)|`

With the Help of Pythagorean Theorem and Substitution Property

`c^2=a^2+b^2\\\\AB^2=AC^2+BC^2\\\\AB^2=(x_(2)-x_(1))^2+(y_(2)-y_(1))^2\\\\AB=\pm \sqrt{(x_(2)-x_(1))^2+(y_(2)-y_(1))^2}`

As, Distance can't be negative

So,
`AB= \sqrt{(x_(2)-x_(1))^2+(y_(2)-y_(1))^2}`

⇒ Here, a and b should be

`a=|y_(2)-y_(1)|,b=|x_(2)-x_(1)|`

Option C:→→ Eowyn didn't correctly derive the distance formula. She drew in her auxiliary segments correctly, but she didn't define a, b, and c correctly in order to apply them in the Pythagorean Theorem.