Question 29004
A rhombus is really an isosceles triangle plus its reflection on its base.
The diagonals meet at right angles. They bisect eachother
sides are
{{{12^2 + 5^2 = s^2}}}
{{{144 + 25 + s^2}}}
{{{169 = s^2}}}
{{{s = 13}}}
draw the figure, dropping an altitude from the 24 diagonal- call it a
[1] {{{x^2 + a^2 = 13^2}}}
[2] {{{24^2 - (13 + x)^2 = a^2}}}
substitute a^2 in [2] for a^2 in [1]
{{{x^2 + 24^2 -(13 +x)^2 = 13^2}}}
{{{x^2 + 24^2 - 13^2 -26*x -x^2 = 13^2}}}
cancel both x^2 and multiply both sides by -1
{{{-24^2 +13^2 + 26*x = -13^2}}}
add +24^2 to both sides
subtract 13^2 from both sides
{{{ 26*x = 24^2 - 2*13^2}}}
{{{ 26*x = 2^2*12^2 - 2*13^2}}}
{{{ 13*x = 2*12^2 - 13^2}}}
{{{ 13*x = 288 - 169}}}
{{{13*x = 119}}}
{{{x = 9.15}}}
now solve for a
{{{a^2 = 13^2 -(9.15)^2}}}
{{{a^2 = 169 - 83.72}}}
{{{a^2 = 85.27}}}
{{{a = 9.23}}}