Mathematics question

Question

A point P lies inside square ABCD. The lines P A, P B, P C, and P D divide the square

into four triangles. Three of the triangles have areas 20, 30, and 45. How many possibilities

are there for the area of the fourth triangle?

Answer 1

Same answer as Roger but different steps

correction, needed to divide b² by 2 to have equations make sense

See that opposite triangle heights add up to b=base

for instance h1+h3=b

those opposite areas added are A1+A3=1/2(b*h1+b*h3)=b/2(h1+h3)=(b²)/2

do same with A2 and unknown A4, A2+A4=(b²)/2 implies A4=(b²)/2-A2=(A1+A3)-A2

there are three combinations of A1+A3, and A2

30+45=75 ,,,,A4=75-20=55

20+45=65,,,A4=65-30=35

20+30=50,,,,A4=50-45=5

Answer 2

Draw and lable a diagram!

Label two opposite triangles 20 and 45. Their area equals one-half the area of the square. (Prove that). Therefore the fourth triangle and the 30 triangle equal one-half of the square. How large is the fourth triangle?

Now the same can be done with the 30 and 45 in opposition.

You should get the possible areas of the square are either 130 or 150.