Question

Rhombus ABCD has perimeter 148, and one of its diagonals has length 24. How long is the other diagonal?

Answer 1

70

Explanation:

The four sides of a rhombus all have equal length, so if the perimeter is 148, then each side has length 148/4 = 37. Also, the diagonals of a rhombus bisect each other at right angles, so the diagonal of length 24 is cut into two pieces of length 12. We can show this information in a diagram (shown below.)

Applying the Pythagorean Theorem to any of the four right triangles in our diagram, we have

12² + x² = 37².

Solving this equation for positive x, we get x = √37² - 12² = √1369 - 144 = √1225 = 35. The length of the long diagonal is x + x = 70.

Answer 2

70 units.

Explanation:

The 4 sides of a rhombus are equal so each side has length 148 /4

= 37 units. The diagonals bisect each other at right angles so when we draw the 2 diagonals we have 4 congruent right-angled triangles.

So consider one of these triangles:

The hypotenuse is 37 units and one of the legs = 1/2 * 24 = 12 units ( half of the diagonal of length 24).

So the other leg is found using Pythagoras:

x^2 = 37^2 - 12^2 = 1225

x = 35.

This is half of the other diagonal so the latter = 70 units.