Question

select the correct answer from each dropdown menu.△abc has vertices of a(-2,5), b(-4,-2), and c(3,-4).the length of ab is square root of 53. the length of ac is square root of 106. the length of bc is . therefore, the triangle is .options for bc length: square root of 106, square root of 53, square root of 26, square root of 85

Explanation:

Step1: Calculate length of BC

Use distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
For B(-4,-2) and C(3,-4):
$\sqrt{(3-(-4))^2+(-4-(-2))^2}=\sqrt{7^2+(-2)^2}=\sqrt{49+4}=\sqrt{53}$

Step2: Classify the triangle

Check side lengths: $AB=\sqrt{53}$, $BC=\sqrt{53}$, $AC=\sqrt{106}$.
Verify Pythagorean theorem: $(\sqrt{53})^2+(\sqrt{53})^2=53+53=106=(\sqrt{106})^2$

Answer:

The length of BC is square root of 53.
Therefore, the triangle is an isosceles right triangle.