Find the dimensions of the rectangle.

Question

 The diagonals of a rectangle are 8 units long and intersects at a 60 degree angle. Find the dimensions of the rectangle.

Answer 1

the diagonals of a rectangle bisect each other

the rectangle I drew has horizontal length greater than vertical width

that said, the 60º angles are the left and right angles at the intersection of the diagonals as opposed to the top and bottom angles at the intersection

you have two triangles that are congruent on the left and on the right

take the triangle on the right

the 2 half-diagonals are each 4 units in length

the other 2 angles of this triangle, on the top and on the bottom, are each 60º because you have an isosceles triangle and the angles opposite the equal sides are equal; 180-60=120 and 120/2=60

so now you see that you have an equilateral triangle with all sides measuring 4 units-that includes the width of the rectangle

the width is 4 units and now we find the length

the length, the width, and the diagonal form a right triangle

the short leg is 4 units and the hypotenuse is 8 units

using the Pythagorean Theorem...

8²=length²+4²

64=length²+16

64-16=length²

48=length²

length=√48=√16*3=4√3

length=4√3 units

width=4 units