Question

Rhombus ADEF is inscribed into a triangle ABC so that they share angle A and the vertex E lies on the side BC . What is the length of the side of the rhombus if AB=c, and AC=b.

Answer 1

The length of the side of the rhombus inscribed in an equilateral triangle ABC with sides AB = c and AC = b is c, provided that AB = BC = AC by the properties of rhombuses and equilateral triangles.

Step-by-step explanation:

To solve for the length of the side of the rhombus inscribed in triangle ABC, we will use the properties of rhombuses and triangles. Given that triangle ABC has equal sides AB and BC, which we denote as r, and that rhombus ADEF shares angle A and vertex E lies on side BC, we can say AB = BC = AC because the rhombus's sides are all equal in length due to its properties.

Since AC is given as b and AB is given as c, we can conclude that b = c if the rhombus can perfectly inscribe within the triangle, implying that triangle ABC is equilateral. Therefore, the length of the side of the rhombus, which is the same as the length of the sides of triangle ABC, would be c.

Answer 2

The length of the rhombus is =
`(bc)/(b+c)cm`

Step-by-step explanation:

It is given that the Rhombus ADEF is inscribed into a triangle ABC so that they share angle A and the vertex E lies on the side BC.Then,

AE is the angle bisector of ∠A, so divides the sides of the triangle into a proportion:

`(BE)/(CE)=(BA)/(AC)=(c)/(b)`

⇒
`(BE)/(CE)=(c)/(b)`

⇒
`(BE)/(BC)=(c)/(c+b)`

Now, ΔDBE is similar to ΔABC, then

DE=
`((BE)/(BC)){*}AC`

=
`((c)/(c+b)){*}b`

=
`(bc)/(b+c)cm`

Thus, the length of the rhombus is =
`(bc)/(b+c)cm`