Answer
No, this is not a right triangle.
Work Step by Step
The distance between A and B:
$$\sqrt{(-2-1)^2+(-5-7)^2}$$
$$\sqrt{9+144}$$
$$\sqrt{153}$$
The distance between B and C:
$$\sqrt{(1-3))^2+(7-15)^2}$$
$$\sqrt{4+64}$$
$$\sqrt{68}$$
The distance between C and D:
$$\sqrt{(-2-3)^2+(-5-15)^2}$$
$$\sqrt{25+400}$$
$$\sqrt{425}$$
Now, check that these three points form a right triangle using the Pythagorean Theorem.
$$153+68=425$$
$$221\ne425$$
No, this is not a right triangle.