Question

Joey is building a frame for a sandbox. The sandbox is going to be a quadrilateral that has the lengths shown.

A rectangle is shown. The length of the top and bottom sides are 8 feet, and the length of the left and right sides are 12 feet. A diagonal that is 14 feet long is drawn from the bottom one corner of the rectangle to the other corner of the rectangle. Points X and C are opposite to the diagonal.

If the diagonal of the sandbox measures 14 feet, which best describes the shape of the sandbox?

a rectangle, because angle C is a right angle
a rectangle, because angle C and angle X are congruent
a quadrilateral, because angle C and angle X are acute
a quadrilateral, because angle C and angle X are obtuse

Answer 1

Choose: "a quadrilateral, because angle C and angle X are acute."

Reason: If it were a rectangle the diagonal would be sqrt(8^2+12^2)=sqrt(208)≈14.422 ft, not 14 ft, so it is not a rectangle. Split the shape along the 14-ft diagonal: each triangle has sides 8, 12, 14. By the law of cosines the angle between the 8-ft and 12-ft sides satisfies

cos θ = (8^2+12^2-14^2)/(2·8·12) = 12/192 = 0.0625,

so θ ≈ arccos(0.0625) ≈ 86.4°, which is acute. Both angles opposite the diagonal are the same acute angle.