# Name the type of triangle formed by the points $\mathrm{A}(-5,6), \mathrm{B}(-4,-2)$ and $\mathrm{C}(7,5)$.

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Given:

The points $\mathrm{A}(-5,6), \mathrm{B}(-4,-2)$ and $\mathrm{C}(7,5)$.

To do:

We have to find the type of triangle formed by the given points.

Solution:

We know that,

The distance between the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

The distance between $A(-5,6)$ and $B(-4,-2) is,$A B =\sqrt{(-4+5)^{2}+(-2-6)^{2}}=\sqrt{1^{2}+(-8)^{2}}=\sqrt{1+64}=\sqrt{65}$The distance between$B(-4,-2)$and$C(7,5)$is,$BC =\sqrt{(7+4)^{2}+(5+2)^{2}}=\sqrt{(11)^{2}+(7)^{2}}=\sqrt{121+49}=\sqrt{170}$The distance between$C(7,5)$and$A(-5,6)$is,$CA=\sqrt{(-5-7)^{2}+(6-5)^{2}}=\sqrt{(-12)^{2}+1^{2}}=\sqrt{144+1}=\sqrt{145}AB^2+CA^2=(\sqrt{65})^2+(\sqrt{145})^2=65+145=210BC^2=(\sqrt{170})^2=170$Here,$AB ≠ BC ≠ CA$and$AB^2+CA^2≠BC^2\$

Therefore, the points $\mathrm{A}(-5,6), \mathrm{B}(-4,-2)$ and $\mathrm{C}(7,5)$ form a scalene triangle.

Updated on 10-Oct-2022 13:28:28