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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