## Geoboard Triangle

The horizontal and vertical distances between adjacent points equal 1 unit. Find the area of triangle $ABC$.

Source: Julia Robinson Mathematics Festival

SOLUTION

area of triangle $AEC$ = area of triangle $ABC$ + area of triangle $ADB$ + area of trapezoid $BDCE$
$\frac{1}{2}AE\times EC$ = area of triangle $ABC$+ $\frac{1}{2}AD\times DB+\frac{1}{2}DE\left (DB+EC\right )$
Multiply both sides by 2
$AE\times EC=2\times$ area of triangle $ABC+ AD\times DB+DE\left (DB+EC\right )$
$4\times 3$ = $2\times$ area of triangle $ABC+3\times 2+1\left (2+3\right )$
Simplify
$12=2\times$ area of triangle $ABC+6+5$
$12=2\times$ area of triangle $ABC+11$
$1=2\times$ area of triangle $ABC$
area of triangle $ABC=\frac{1}{2}$
Answer: $\frac{1}{2}$ square unit.