How do I solve this please? Show that if the sides of a right triangle are in an arithmetic sequence, then their ratio is 3:4:5.


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Question Number 199986 by Yakubu last updated on 11/Nov/23

How do I solve this please? Show that if the sides of a right triangle are in an arithmetic sequence, then their ratio is 3:4:5.
	Answered by Frix last updated on 12/Nov/23

	$a^{2} + {(a + k)}^{2} = {(a + 2 k)}^{2} with a, k > 0$ $a^{2} - 2 k a - 3 k^{2} = 0$ $(a + k) (a - 3 k) = 0$ $k = - a impossible$ $k = \frac{a}{3}$ $All triangles with sides a, \frac{4 a}{3}, \frac{5 a}{3} are$ $rectangular and their sides are in an$ $arithmetic sequence and their ratio is$ $3 : 4 : 5$
	Commented by Yakubu last updated on 12/Nov/23

	$t h a n k y o u s i r .$
	Answered by Rasheed.Sindhi last updated on 12/Nov/23
	$AnOther way: Let the required traingle has sides in ratio l:m:n We′ll prove that l:m:n=3:4:5 la,ma,na are sides { (((la)^2 +(ma)^2 =(na)^2 ⇒l^2 +m^2 =n^2 )),((ma=((la+na)/2)⇒m=((l+n)/2))) :} l^2 +(((l+n)/2))^2 =n^2 4l^2 +l^2 +2ln+n^2 =4n^2 5l^2 +2ln=3n^2 ((5l)/(2n))+1=((3n)/(2l)) (5/2)t−(3/(2t))+1=0 5t^2 +2t−3=0 (t+1)(5t−3)=0 t=(l/n)=−1(absurd ∵ l,n∈Z^+ ∣ t=(3/5)⇒(l/n)=(3/5)⇒n=((5l)/3) m=((l+n)/2)⇒2m=l+((5l)/3) ⇒6m=8l⇒(l/m)=(3/4) (l/n)=(3/5) ∧ (l/m)=(3/4)⇒ determinant (((l:m:n=3:4:5))) proved$
	$AnOther way :$ $L e t t h e r e q u i r e d t r a i n g l e h a s$ $s i d e s i n r a t i o l : m : n$ ${W e}^{'} l l p r o v e t h a t l : m : n = 3 : 4 : 5$ $l a, m a, n a a r e s i d e s$ ${\begin{cases} {(l a)}^{2} + {(m a)}^{2} = {(n a)}^{2} \Rightarrow l^{2} + m^{2} = n^{2} \\ m a = \frac{l a + n a}{2} \Rightarrow m = \frac{l + n}{2} \end{cases}$ $l^{2} + {(\frac{l + n}{2})}^{2} = n^{2}$ $4 l^{2} + l^{2} + 2 l n + n^{2} = 4 n^{2}$ $5 l^{2} + 2 l n = 3 n^{2}$ $\frac{5 l}{2 n} + 1 = \frac{3 n}{2 l}$ $\frac{5}{2} t - \frac{3}{2 t} + 1 = 0$ $5 t^{2} + 2 t - 3 = 0$ $(t + 1) (5 t - 3) = 0$ $t = \frac{l}{n} = - 1 (a b s u r d ∵ l, n \in Z^{+} ∣$ $t = \frac{3}{5} \Rightarrow \frac{l}{n} = \frac{3}{5} \Rightarrow n = \frac{5 l}{3}$ $m = \frac{l + n}{2} \Rightarrow 2 m = l + \frac{5 l}{3}$ $\Rightarrow 6 m = 8 l \Rightarrow \frac{l}{m} = \frac{3}{4}$ $\frac{l}{n} = \frac{3}{5} \land \frac{l}{m} = \frac{3}{4} \Rightarrow \begin{matrix} l : m : n = 3 : 4 : 5 \end{matrix}$ $p r o v e d$
	Commented by Yakubu last updated on 12/Nov/23

	$t h a n k y o u s i r$
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