Triangle ABC is symmetric about the y-axis. Point A is located at (-4,0), and AB is the longest side of ABC. If the perimeter of ABC is 18, what is the area?

- 3√2
- 6
- 6√2
- 12
- 24

## Step #1

With coordinate geometry questions in which there is no coordinate plane, always make sure to draw the plane and any diagram (with practice, you’ll get quick at making a rudimentary version). Trying to visualize the information is simply much more difficult.

## Step #2

Once you’ve drawn all the information, which should include Point A and B, then the next steps should be easier. By the way, point B is located at (4,0). Remember, ABC is symmetric about the y-axis, meaning that exactly half of the triangle is on the left side of the y-axis; the other half is on the right side. The only way to arrange such a triangle is to have AB as a horizontal line, which forms the base of the triangle.

If you don’t believe me try drawing a triangle in which AB is not a horizontal line and the triangle is symmetric about the y-axis.

## Step #3

At this point, you may have also noticed that the triangle is an isosceles triangle (equal parts on both sides of the axis). Therefore the top of triangle is located on the y-axis. Whether it is above line AB or below doesn’t matter, as we are only looking for the area of ABC (the height will be constant—the distance point C is from side AB).

## Step #4

Because we have an isosceles triangle, BC and AC must both equal 5. Remember the perimeter is 18. AB accounts for 8, leaving 10 for both sides. Splitting the triangle in half (basically each side of the triangle), we have two 3:4:5 right triangles. Therefore the height is 3, and the area of the triangle is 12, Answer (D).

## Bonus Step:

This problem could have been even more difficult had I not provided the information that AB is the longest side. In that case, we still would have had the exact same triangle. Point B and Point C could be switched, but that detail is moot because the dimensions of the triangle would have been exactly the same.

Dear Chris,

I am from Nepal. I am preparing for GRE. I am going through Barron’s book. I regard myself as a layman in mathematics since almost all of the things I learned about mathematics during my school days elude me now. Therefore, what I am going to ask may be so simple that I should in fact feel ashamed to ask it. Despite this possibility, let me ask one question to ask. From the fact that the triangle ABC is symmetric about y-axis , we can easily know that the base of the triangle AB is 8(4+4). From this fact, we can also deduce that the sum of the remaining two sides is 10. But I am not clear about how we can say with certitude that the two sides are equal, with each side being 5. This is my first question. Another question I would like to ask is: let’s suppose that they are equal, meaning that the triangle is isosceles triangle. What is the quickest way of knowing that the point C lies at the (0, 3) coordinates ?

thank you mis lele, would please explain more about congruent triangle and its property in deatails..

Anik,

A right triangle that is congruent is always a 45:45:90 triangle. These triangles have the property that the two shorter sides are always x and x, and the longest side (the hypotenuse) is xrt2.

In general, congruent triangles always share two sides that have equal angle measures.

Hope that helps!