In the figure, given coordinates of point $$A$$ is $$(3,4)$$ in the rectangular coordinates. Let point $$P$$ is a dynamic point on the positive $$x$$ axis such that $$ riangle AOP$$ is an isosceles triangle. How many such triangle could be formed?

Question

In the figure, given coordinates of point $$A$$ is $$(3,4)$$ in the rectangular coordinates. Let point $$P$$ is a dynamic point on the positive $$x$$ axis such that $$	riangle AOP$$ is an isosceles triangle. How many such triangle could be formed?

Uniteasy Admin · Accepted Answer

Moving the point $$P$$ along the $$x$$ axis finds 3 isosceles triangles could be formed.
Since point A is a fixed point, using the distance formula between two points gives
$$AO = \sqrt{3^3+4^2}  $$
$$\cos \angle AOP = \dfrac{3}{5}  $$
1. When $$AP = OP$$
$$OP = \dfrac{OD}{\cos \angle AOP}  $$

$$ = \dfrac{\dfrac{AO}{2}    }{\cos \angle AOP}  $$

$$= \dfrac{2.5}{\dfrac{3}{5}  } = \dfrac{5}{2}\cdot \dfrac{5}{3}=\dfrac{25}{6}        $$
The coodinates of $$P$$ is $$(\dfrac{25}{6} ,0)$$
2. When $$OP  = OA = 5$$
The coodinates of $$P$$ is $$(5,0)$$
3. When OP = AP
The coodinates of $$P$$ is $$(6,0)$$

Multiple Choice Question (MCQ)

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