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. the length of ac is
the length of bc is. therefore, the triangle is
(options in dropdown: square root of 85, square root of 26, square root of 106, square root of 53)

Explanation:

Step1: Calculate length of AB

Use distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$AB=\sqrt{(-4-(-2))^2+(-2-5)^2}=\sqrt{(-2)^2+(-7)^2}=\sqrt{4+49}=\sqrt{53}$

Step2: Calculate length of AC

$AC=\sqrt{(3-(-2))^2+(-4-5)^2}=\sqrt{(5)^2+(-9)^2}=\sqrt{25+81}=\sqrt{106}$

Step3: Calculate length of BC

$BC=\sqrt{(3-(-4))^2+(-4-(-2))^2}=\sqrt{(7)^2+(-2)^2}=\sqrt{49+4}=\sqrt{53}$

Step4: Classify the triangle

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

Answer:

• The length of AB is: square root of 53
• The length of AC is: square root of 106
• The length of BC is: square root of 53
• Therefore, the triangle is: an isosceles right triangle