Question

a manufacturer cuts a piece of metal for a microscope. the resulting piece of metal can be represented in a coordinate plane by a triangle with vertices ( a(0, 0) ), ( b(3, 8) ), and ( c(6, 0) ). one unit in the coordinate plane represents one millimeter. prove that ( \triangle abc ) is isosceles.

find the exact length of each side.

( ab = square ) mm

( bc = square ) mm

( ac = square ) mm

because ( square cong square ), ( \triangle abc ) is isosceles.

Explanation:

Step1: Recall the distance formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Calculate the length of \(AB\)

For points \(A(0,0)\) and \(B(3,8)\), substitute \(x_1 = 0,y_1 = 0,x_2=3,y_2 = 8\) into the distance formula:
\(AB=\sqrt{(3 - 0)^2+(8 - 0)^2}=\sqrt{3^2 + 8^2}=\sqrt{9 + 64}=\sqrt{73}\)

Step3: Calculate the length of \(BC\)

For points \(B(3,8)\) and \(C(6,0)\), substitute \(x_1 = 3,y_1 = 8,x_2=6,y_2 = 0\) into the distance formula:
\(BC=\sqrt{(6 - 3)^2+(0 - 8)^2}=\sqrt{3^2+(- 8)^2}=\sqrt{9 + 64}=\sqrt{73}\)

Step4: Calculate the length of \(AC\)

For points \(A(0,0)\) and \(C(6,0)\), substitute \(x_1 = 0,y_1 = 0,x_2=6,y_2 = 0\) into the distance formula:
\(AC=\sqrt{(6 - 0)^2+(0 - 0)^2}=\sqrt{6^2+0^2}=\sqrt{36}=6\)

Answer:

\(AB=\boldsymbol{\sqrt{73}}\) mm, \(BC=\boldsymbol{\sqrt{73}}\) mm, \(AC = \boldsymbol{6}\) mm. Since \(AB = BC\), \(\triangle ABC\) is isosceles.