Coordinates of D: A(-2,2), B(4,4) and C(5,2) are the vertices of a triangle...

[cired2002]

The points A(-2,2), B(4,4) and C(5,2) are the vertices of a triangle. The perpendicular bisector of AB and the line through A parallel to BC intersect at the point D. Find the area of the quadrilateral ABCD. pls help!

 

[tkhunny]

Have you considered constructing it all?

Plot the points.
Draw the lines.
Construct the perpendicular bisector.
etc...

Go!

 

[mmm4444bot]

The points A(-2,2), B(4,4) and C(5,2) are the vertices of a triangle. The perpendicular bisector of AB and the line through A parallel to BC intersect at the point D. Find the area of the quadrilateral ABCD.

What did you try? How far did you get? Where are you stuck?

There are different ways to solve this exercise. I'm thinking of a way that uses Heron's Formula, for finding the area of a triangle when you know its three side lengths.

If you're not sure how to begin, try tkhunny's suggestion: draw a rough sketch, and find an equation for AB's perpendicular bisector and for the line through A (parallel to BC). Use these equations, to determine the coordinates at their intersection point (D).

Now, from your diagram, you can see that quadrilateral ABCD can be decomposed into two triangles: ABC ACD.

You can calculate the area of triangle ABC directly from its base and height. You can use Heron's Formula, to calculate the area of triangle ACD.

Sum the areas. :cool:

If you haven't learned Heron's Formula, yet, let us know.

If you've forgotten how to write equations of lines, or how to find their intersection points, or how to use the Distance Formula, let us know.

If you've been thinking of a different approach, let us know.

 

[cired2002]

What did you try? How far did you get? Where are you stuck?

There are different ways to solve this exercise. I'm thinking of a way that uses Heron's Formula, for finding the area of a triangle when you know its three side lengths.

If you're not sure how to begin, try tkhunny's suggestion: draw a rough sketch, and find an equation for AB's perpendicular bisector and for the line through A (parallel to BC). Use these equations, to determine the coordinates at their intersection point (D).

AB's perpendicular bisector intersects side AC. Let's name this intersection point E. Determine the coordinates of point E.

Now, from your diagram, you can see that quadrilateral ABCD can be decomposed into three triangles: ABC, ADE, and ACE.

You can calculate the area of triangle ABC directly from its base and height. You can use Heron's Formula, to calculate the areas of triangles ADE and ACE.

Sum these three areas. :cool:

If you haven't learned Heron's Formula, yet, let us know.

If you've forgotten how to write equations of lines, or how to find their intersection points, or how to use the Distance Formula, let us know.

If you've been thinking of a different approach, let us know.

I have tried constructing the points. I have form the eqn of the perpendicular line (BD) and AD. I know heron formula. I tried solving them simultaneously but I got the coordinates of A instead

 

[mmm4444bot]

… I have [formed] the eqn of the perpendicular line (BD) …

Is that a typo? The perpendicular line bisects side AB; it does not pass through point B.

What is your equation for the perpendicular bisector of AB?

I wrote it, using these steps:

find slope of AB
find midpoint of AB
use Point-Slope Formula​

The perpendicular bisector's slope is the negative reciprocal of AB's slope.


Also, what is your equation for the line passing through AD?