GIVEN
A triangle DOG with the coordinates of the vertices given to be: D(1, 9), O(7, 9), and G(4, 2).
SOLUTION METHOD
To correctly classify the triangle, we need to get the lengths of each side of the triangle.
The formula to calculate the length of a line between two given points is given to be:
![AB=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}](https://img.qammunity.org/2023/formulas/mathematics/college/kvsd6z78gv9ailf7xas619os36u1wah5zo.png)
Distance of Line DO:

Therefore, the length is given to be:
![\begin{gathered} DO=\sqrt[]{(7-1)^2+(9-9)^2} \\ DO=\sqrt[]{6^2+0^2} \\ DO=\sqrt[]{36} \\ DO=6 \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/wa9431waf6lw52iaslmqkhho06g7vopmz1.png)
Distance of Line OG:

Therefore, the length is given to be:
![\begin{gathered} OG=\sqrt[]{(4-7)^2+(2-9)^2} \\ OG=\sqrt[]{(-3)^2+(-7)^2} \\ OG=\sqrt[]{9+49} \\ OG=\sqrt[]{58} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/m85bhv6en3frhey7b5m6m4u41v95z9qhy3.png)
Distance of Line DG:

Therefore, the length is given to be:
![\begin{gathered} DG=\sqrt[]{(4-1)^2+(2-9)^2} \\ DG=\sqrt[]{3^2+(-7)^2} \\ DG=\sqrt[]{9+49} \\ DG=\sqrt[]{58} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/hex6waz2hb1wmnw00005txdu5clokmozrq.png)
The lengths of the sides are:
![\begin{gathered} DO=6 \\ OG=\sqrt[]{58} \\ DG=\sqrt[]{58} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/9esylji9fhcpenuqb4yyeae2bw34frzgmv.png)
CONCLUSION
Since two of the sides are equal, then the triangle is an ISOSCELES TRIANGLE.