Question

Let o be the origine of co ordinate plane , A,B lie on the upper half plane satisfying OA=OB. If line OA has slope 1, line OB has slope -7, what is the slope of line AB ?

Answer 1

Answer on Question #82066 – Math – Geometry

Question

Let $o$ be the origin of coordinate plane, $A$, $B$ lie on the upper half plane satisfying $OA = OB$. If line $OA$ has slope 1, line $OB$ has slope -7, what is the slope of line $AB$?

Solution

Let $A$ have coordinates $(x, y)$. $y = x$ because line $OA$ has slope 1 which equals $y / x$.

Let $B$ have coordinates $(-a, b)$. $b = 7a$ because line $OB$ has slope -7 which equals $b / a$ (line placed in left upper half plane)

By using the Pythagorean theorem calculating length of line OA:

$$\operatorname{sqrt}(x^2 + x^2) = x * \operatorname{sqrt}(2)$$

We know that the line $OB$ have the same length so we can set correspondence between $x$ and $a$:

Length $OB$ equals $\operatorname{sqrt}(7a)^2 + a^2 = x * \operatorname{sqrt}(2) \Leftrightarrow (7a)^2 + a^2 = 2 * x^2$

$$\Leftrightarrow 25*a^2 = x^2 \Leftrightarrow 5a = x$$

So what we've got? By using this correspondence we know that coordinates of $A$ are $(5a, 5a)$.

Coordinates of $B$ are $(-a, 7a)$. Let's enter a point $C$ with coordinates $(-a, 5a)$. Line $AC$ parallel to the $x$-axis because points $A$ and $C$ have the same ordinates. That means that the tangent of angle $BAC$ equals minus slope of line $AB$ because angle $BAC$ adjacent to the angle which defines the slope of line $AB$ (180 – angle $BAC$) so as we know $\tan(180 - \alpha) = -\tan(\alpha)$

Tangent of angle $BAC$ equals $BC / AC = (7a - 5a) / (5a - (-a)) = 2a / 6a = 1 / 3$ hence slope of line $AB$ equals $-1 / 3$

Answer: $-1 / 3$

Answer provided by https://www.AssignmentExpert.com