The three sides of a right-angled triangle ar | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the sides of the right-angled triangle be $a, ar, ar^2$ where $ar^2$ is the hypotenuse.By Pythagoras theorem,$(ar^2)^2 = a^2 + (ar)^2$.Dividin

Vedclass · Accepted Answer

$\sqrt{\frac{\sqrt{5}+1}{2}}$ and $\sqrt{\frac{\sqrt{5}-1}{2}}$

Vedclass · Answer

(B) Let the sides of the right-angled triangle be $a, ar, ar^2$ where $ar^2$ is the hypotenuse.By Pythagoras theorem,$(ar^2)^2 = a^2 + (ar)^2$.Dividing by $a^2$ (assuming $a 
eq 0$),we get $r^4 = 1 + r^2$,which implies $r^4 - r^2 - 1 = 0$.Solving for $r^2$ using the quadratic formula,$r^2 = \frac{1 \pm \sqrt{1 - 4(1)(-1)}}{2} = \frac{1 + \sqrt{5}}{2}$ (since $r^2 > 0$).Thus,$r = \sqrt{\frac{\sqrt{5}+1}{2}}$.In the triangle,$	an \alpha = \frac{	ext{opposite}}{	ext{adjacent}} = \frac{ar}{a} = r = \sqrt{\frac{\sqrt{5}+1}{2}}$.Similarly,$	an \beta = \frac{a}{ar} = \frac{1}{r} = \frac{1}{\sqrt{\frac{\sqrt{5}+1}{2}}} = \sqrt{\frac{2}{\sqrt{5}+1}} = \sqrt{\frac{2(\sqrt{5}-1)}{5-1}} = \sqrt{\frac{\sqrt{5}-1}{2}}$.Therefore,the values are $\sqrt{\frac{\sqrt{5}+1}{2}}$ and $\sqrt{\frac{\sqrt{5}-1}{2}}$.

The three sides of a right-angled triangle are in $GP$ (geometric progression). If the two acute angles are $\alpha$ and $\beta$,then $\tan \alpha$ and $\tan \beta$ are

Similar Questions

Six positive numbers are in $GP$,such that their product is $1000$. If the fourth term is $1$,then the last term is

The geometric mean of the first $n$ terms of the sequence $a, ar, ar^2, \dots$ is:

If the $10^{th}$ term of a geometric progression is $9$ and the $4^{th}$ term is $4$,then its $7^{th}$ term is

In a geometric progression with $n$ terms and a common ratio of $3$,the sum of the $n$ terms is $364$ and the last term is $243$. Find the value of $n$.

The least positive integer $n$ such that $1 - \frac{2}{3} - \frac{2}{3^2} - \dots - \frac{2}{3^{n-1}} < \frac{1}{100}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module