In triangle ABC,if r_1=3, r_2=10 and r_3=15,t | vedclass

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

Question

(C) We know the relation between the exradii $r_1, r_2, r_3$ and the circumradius $R$ is given by $\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \fr

Vedclass · Accepted Answer

$\frac{13}{2}$

Vedclass · Answer

(C) We know the relation between the exradii $r_1, r_2, r_3$ and the circumradius $R$ is given by $\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \frac{1}{r}$,where $r$ is the inradius. Also,the area of the triangle $\Delta = \sqrt{r r_1 r_2 r_3}$. Given $r_1=3, r_2=10, r_3=15$. $\frac{1}{r} = \frac{1}{3} + \frac{1}{10} + \frac{1}{15} = \frac{10+3+2}{30} = \frac{15}{30} = \frac{1}{2}$. So,$r=2$. Now,$\Delta = \sqrt{2 	imes 3 	imes 10 	imes 15} = \sqrt{900} = 30$. We also have the formula $\Delta = rs$,where $s$ is the semi-perimeter. $30 = 2s \implies s = 15$. Using the formula $r_1 = \frac{\Delta}{s-a}$,we get $3 = \frac{30}{15-a} \implies 15-a = 10 \implies a = 5$. Similarly,$r_2 = \frac{\Delta}{s-b} \implies 10 = \frac{30}{15-b} \implies 15-b = 3 \implies b = 12$. And $r_3 = \frac{\Delta}{s-c} \implies 15 = \frac{30}{15-c} \implies 15-c = 2 \implies c = 13$. Since $a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169 = 13^2 = c^2$,the triangle is a right-angled triangle. For a right-angled triangle,the circumradius $R = \frac{c}{2} = \frac{13}{2} = 6.5$.

In $\triangle ABC$,if $r_1=3, r_2=10$ and $r_3=15$,then $R=$

Similar Questions

$PQR$ is an isosceles triangle with $PQ=PR$. If the radius of the circumcircle of $\triangle PQR$ is equal to the length of $PQ$,then $\angle P=$ (in $^{\circ}$)

If the vertices of a triangle are $(2, -2)$,$(8, -2)$,and $(8, 6)$,what is the excenter opposite to the vertex $(2, -2)$?

In $\triangle ABC$,if $b=2, c=\sqrt{3}$,and $A=30^{\circ}$,then its inradius $r=$

If $ABC$ is an isosceles triangle with base $BC$,then $rr_1=$

If the lengths of the sides of a triangle are $15, 20, 25$ units,find the circumradius of the triangle. (in $units$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module