To solve the problem, let's first visualize the square ABCD with side length 16, and place it on a coordinate plane:
- Assume point A=(0,16), B=(16,16), C=(16,0), and D=(0,0).
Points E and F divide side CD into three equal segments:
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Since CD=16, each segment CE=EF=FD=316≈5.33.
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The coordinates for E and F are:
- E=(16−316,0)=(332,0)
- F=(16−2×316,0)=(316,0)
Step 1: Find Equations of Lines AE and BF
Line AE:
- Point A=(0,16) and E=(332,0).
- Slope of line AE is:
slope of AE=332−00−16=−33216=−3216⋅3=−23
- The equation of line AE in point-slope form is:
y−16=−23x⟹y=−23x+16
Line BF:
- Point B=(16,16) and F=(316,0).
- Slope of line BF is:
slope of BF=316−160−16=316−16−16=316−48−16=−32−16⋅3=3248=23
- The equation of line BF in point-slope form is:
y−16=23(x−16)⟹y=23x−8
Step 2: Find Coordinates of Point M
To find the intersection M of lines AE and BF, set their equations equal:
−23x+16=23x−8
Combine like terms:
16+8=23x+23x⟹24=3x⟹x=8
Substitute x=8 back into either line equation to find y:
y=−23⋅8+16=−12+16=4
Thus, point M=(8,4).
Step 3: Calculate the Area of Triangle MAB
The vertices of triangle MAB are M=(8,4), A=(0,16), and B=(16,16).
The area of a triangle given by coordinates (x1,y1), (x2,y2), (x3,y3) is:
Area=21∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣
Substitute the coordinates:
Area=21∣8(16−16)+0(16−4)+16(4−16)∣
Simplify:
Area=21∣0+0+16(−12)∣=21∣−192∣=2192=96
Conclusion
The area of triangle MAB is 96.