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**

Did you like the problem? Would you like to have these problems and solutions delivered to your inbox as I publish them? Click here to subscribe

Want to prepare for AMC 8? Working out the problems by yourself is the only way to prepare for it. So get your brains working and **write your answers in the comments section below.**

__#Competition__, #Math #Problems #OnlineMathTutoring, #MiddleSchoolMathEnrichment, #OnlineMathEnrichmentPrograms #AMC8 #AMC8Tutor #OnlineMathTutor #ProblemOfTheWeek

**Previous Week's Problem**

**Next Week's Problem**

## Comments