In a right triangle ABC,the incircle touches | vedclass

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

Question

(D) Let the incircle have center $O$ and radius $r$. The incircle touches $AC$ at $D$,$BC$ at $E$,and $AB$ at $F$.Since $O$ is the center and $OE \per

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(D) Let the incircle have center $O$ and radius $r$. The incircle touches $AC$ at $D$,$BC$ at $E$,and $AB$ at $F$.Since $O$ is the center and $OE \perp BC$,$OF \perp AB$,and $OE=OF=r$,the quadrilateral $O E B F$ is a square with side length $r$.Thus,$BE = BF = r$.By the property of tangents from an external point to a circle,we have:$AF = AD = 10$$CE = CD = 3$Therefore,the sides of the right-angled $	riangle ABC$ are:$AB = AF + BF = 10 + r$$BC = CE + BE = 3 + r$$AC = AD + DC = 10 + 3 = 13$Using the Pythagorean theorem,$AB^2 + BC^2 = AC^2$:$(10 + r)^2 + (3 + r)^2 = 13^2$$100 + 20r + r^2 + 9 + 6r + r^2 = 169$$2r^2 + 26r + 109 = 169$$2r^2 + 26r - 60 = 0$$r^2 + 13r - 30 = 0$$(r + 15)(r - 2) = 0$Since $r > 0$,we have $r = 2$.Thus,the inradius is $2$.

In a right $\triangle ABC$,the incircle touches the hypotenuse $AC$ at $D$. If $AD=10$ and $DC=3$,the inradius of $\triangle ABC$ is

Similar Questions

In $\triangle ABC$,if $a:b:c = 4:5:6$,then the ratio of the circumradius to its inradius is

Let $ABC$ be an equilateral triangle with side length $a$. Let $R$ and $r$ denote the radii of the circumcircle and the incircle of triangle $ABC$ respectively. Then,as a function of $a$,the ratio $\frac{R}{r}$

In a triangle $ABC$,$a : b : c = 4 : 5 : 6$. The ratio of the radius of the circumcircle to that of the incircle is

In $\triangle ABC$,with usual notation,observe the two statements given below :
$(I)$ $r r_1 r_2 r_3 = \Delta^2$
$(II)$ $r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2$
Which of the following is correct?

If the sides of a triangle are $13, 14, 15$,then the radius of its incircle is

Vedclass Test Series

Exam Paper Generator

Online Exam Module