Question 1019325
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The medians of a right triangle that are drawn from the vertices of the acute angles have lengths of 2 square root 13 and square root 73. Find the lengths of the hypotenuse.
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<U>Answer</U>. The length of the hypotenuse is 10 units.


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<U>Solution</U>

We will use this property of a median, which is valid for any triangle:

  In a triangle with the sides a, b and c, the median drawn to the side c has the length {{{m[c]}}} = {{{sqrt((2a^2 + 2b^2 - c^2)/2)}}}.

See the lesson <A HREF=The length of a median of a triangle>The length of a median of a triangle</A> in this site.

Next, let us apply the property to a right-angled triangle, whose legs are a and b units long and the hypotenuse is c units long.

For the medians  {{{m[a]}}}  and  {{{m[b]}}}  drawn to the legs a and b respectively, we will have

{{{m[a]^2}}} = {{{(2b^2 + 2c^2 - a^2)/4}}},  {{{m[b]^2}}} = {{{(2a^2 + 2c^2 - b^2)/4}}}. 

Therefore,

{{{m[a]^2}}} + {{{m[b]^2}}} = {{{(2b^2 + 2c^2 - a^2)/4}}} + {{{(2a^2 + 2c^2 - b^2)/4}}} = {{{(2b^2 + 2c^2 - a^2 + 2a^2 + 2c^2 - b^2)/4}}} = {{{(a^2 + b^2 + 4c^2)/4}}}.

Since for the right-angled triangle {{{a^2 + b^2}}} = {{{c^2}}}, you can rewrite the above equality in the form

{{{m[a]^2}}} + {{{m[b]^2}}} = {{{(c^2 + 4c^2)/4}}} = {{{(5/4)*c^2}}}.

Now substitute the given data  {{{m[a]}}} = {{{2*sqrt(13)}}}  and  {{{m[b]}}} = {{{sqrt(73)}}}. You will get

{{{(5/4)c^2}}} = {{{4*13 + 73}}} = {{{125}}}.

It implies  {{{c^2}}} = {{{(4/5)*125}}} = {{{100}}}.

Hence, c = 10.

The problem is solved.
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