Question

in the figure p is the incenter of isosceles rst what type of triangle is rpt? Thank you in advance .

Answer 1

Isosceles triangles mean that 2 sides of the triangle are congruent. Since triangle RST is isosceles and P is its incenter, that means that PT and PR must be congruent, also. PRT is an isosceles triangle.

Answer 2

`\Delta \text{RPT is an isosceles triangle}`

Explanation:

As in the triangle RST,

`RS=ST\ \ \ \Rightarrow m\angle SRT=m\angle STR`

Because if in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.

Incenter is obtained by the Intersection of a triangle's three angle bisectors and it is equidistant from the three sides of the triangle.

So, RP and TP are the angle bisectors of ∠R and ∠T respectively.

`\Rightarrow (m\angle SRT)/(2)=(m\angle STR)/(2)`

`\Rightarrow m\angle PRT=m\angle PTR`

`\Rightarrow PR=PT`

`\Rightarrow \Delta \text{RPT is an isosceles triangle}`