Question #6b0b4

1 Answer
Sep 23, 2015

Grilletti - The key to solving a problem like this is to ALWAYS sketch a picture.

Explanation:

Here is sketch of the problem:

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There are exactly two lines that will pass through the point (-3, -4) and that will also be 5 units away from the point (2,1).

Drawing a circle with radius = 5 helps to visualize the problem.

Notice that both lines #bar(AB) and bar(AB'# are tangent lines to the circle. Also note, #/_ ABC and /_AB'C# are both right angles formed by the tangent lines and the radii of the circle.

Both lines #bar(AB) and bar(AB'# show a length of 5 units. Let's prove that now:

#Delta#ABC and #Delta#AB'C are both right triangles that are congruent by the theorem HL (hypotenuse-leg). We can calculate the length of the hypotenuse by using the distance formula between two points (I'll leave that for you to prove). The length of #bar(AC)# is #5sqrt2#. Finally, use the Pythagorean Theorem to find the length of #bar(AB) and bar(AB'# which both are equal to 5 units. This implies that the coordinate of point B is (-3, 1) and point B' is (2, -4) which means #bar(AB)# is vertical line and #bar(AB')# is a horizontal line .

Now we have all the information we need to determine the equations of the two lines #bar(AB) and bar(AB'#:

#bar(AB)# is a vertical line #x =-3#

#bar(AB'# is a horizontal line #y =-4#

Hope that helped!