Question

students are trying to find the distance between the two points (9,19) and (-2,15). find the mistake in each students work.
find the mistake for student 3
distance = \\( \sqrt { ( 9 - ( - 2 ) ) ^ { 2 } + ( 19 - 15 ) ^ { 2 } } \\)
= \\( \sqrt { ( 11 ) ^ { 2 } + ( 4 ) ^ { 2 } } \\)
= 11 + 4
= 15
show how to correctly find the distance between the two given points. substitute (9,19) for \\( ( x _ { 1 }, y _ { 1 } ) \\) and (-2,15) for \\( ( x _ { 2 }, y _ { 2 } ) \\) in the distance formula.
distance = \\( \sqrt { ( x _ { 2 } - x _ { 1 } ) ^ { 2 } + ( y _ { 2 } - y _ { 1 } ) ^ { 2 } } \\)
\\( \sqrt { ( - 2 - \square ) ^ { 2 } + ( 15 - \square ) ^ { 2 } } \\)

Explanation:

Step1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 9,y_1 = 10,x_2=-2,y_2 = 15\).

Step2: Analyze the student's error

The student's work: \(d=\sqrt{(9-(-2))^2+(10 - 15)^2}=\sqrt{11^2+(- 5)^2}\). But the student wrote \(\sqrt{11^2+4^2}\) (incorrectly calculated \((10 - 15)^2\) as \(4^2\) instead of \((-5)^2 = 25\)). The main error is that the student did not follow the order of operations. The correct steps are:

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The student added \(11^2\) and \(4^2\) (from wrong calculation of \((10 - 15)\)) and then took the sum \(11 + 4\) instead of first adding \(11^2+(-5)^2\) and then taking the square - root.

Answer:

The student did not do the operations in the correct order. The student needed to square and add before taking the square root. The correct distance formula substitution is \(\sqrt{(-2 - 9)^2+(15 - 10)^2}\)