Question

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are △stu and △hij congruent?

Explanation:

Step1: Calculate the side lengths of \(\triangle STU\)

Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) or counting units on the grid:

• \(ST=\sqrt{(0 - (- 9))^2+(8 - 1)^2}=\sqrt{81 + 49}=\sqrt{130}\) (or count \(9\) units in \(x\) - direction and \(7\) units in \(y\) - direction, then by Pythagoras \(a=\sqrt{9^{2}+7^{2}}=\sqrt{81 + 49}=\sqrt{130}\))
• \(TU = 9\) (counting units along the vertical line from \(T(-9,0)\) to \(U(-9,9)\))
• \(SU=9\) (counting units along the horizontal line from \(S(0,9)\) to \(U(-9,9)\))

Step2: Calculate the side lengths of \(\triangle HIJ\)

• \(HI=\sqrt{(9 - 1)^2+(-9-(-9))^2}=\sqrt{64+0}=8\) (counting units along the horizontal line from \(I(9,-9)\) to \(J(1,-9)\))
• \(IJ=\sqrt{(9 - 1)^2+( - 9-4)^2}=\sqrt{64 + 169}=\sqrt{233}\) (using Pythagoras: \(a=\sqrt{(9 - 1)^{2}+(-9 - 4)^{2}}=\sqrt{64+169}=\sqrt{233}\))
• \(HJ=\sqrt{(1 - 1)^2+(4+9)^2}=13\) (counting units along the vertical line from \(H(1,4)\) to \(J(1,-9)\))

Since the side - lengths of \(\triangle STU\) (\(\sqrt{130},9,9\)) and \(\triangle HIJ\) (\(8,\sqrt{233},13\)) are not equal.

Answer:

\(\triangle STU\) and \(\triangle HIJ\) are not congruent.