Math Competition Problems - Problem of the Week - 17

Updated: Jul 30, 2019

In the figure below, choose point D on BC so that triangle ACD and triangle ABD have equal perimeters. What is the area of triangle ABD? - AMC 8 2017


STEP 1: Find BD such that the perimeter of triangle ACD is equal to the perimeter of triangle ABD.

We have a triangle ABC. Let's draw a line from A to D such that the perimeter of triangle ACD is equal to the perimeter of triangle ABD. And let us call the distance from A to D as d.

Let CD = x and BD = y.


Now we have x+y=5. Let's call this equation 1.

Since the perimeter of triangle ACD is equal to the perimeter of triangle ABD, we have

3+d+x = 4+d+y

3+x = 4+y

x = y+1

Let's call this equation 2.


Now substitute x = y+1 in the first equation of x+y=5. You get

y+1+y = 5

y = 2

and plugging y = 2 into equation 2 we get x = 2+1

and hence x = 3.


Now that we have the distance from B to D. We set out to find the area of triangle ABD.

To find the area of a triangle we its base and height.


STEP 2: Find h the height of the triangle ABD

If we consider the base as BD, the height would be a line drawn from A to BD (or BC) perpendicular to BD.

Let's draw a line AH from A to BC and let's call the distance from A to H as h.

We know that ABC is a right triangle because 3^2 + 4^2 = 5^2. Or we also know that 3,4,5 are Pythagorean triplets.


Now if we look at triangle ABC (the original large triangle), there are 2 ways we can calculate its area.


First we can consider AB to be the base and hence AC would be the height. So the area of triangle ABC = 1/2*base*height = 1/2*4*3=6 sq. units.


We can also see that the area of the triangle ABC can be calculated by using BC as the base and h as the height. Since we know the area of ABC is 6, we have

1/2 * 5 * h = 6

and hence h=12/5.


STEP 3: Calculate the area of triangle ABD

For triangle ABD base is BD=2; and height is AH = 12/5.

Area = 1/2 * 2 * 12/5

Area = 12/5


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