Question

are △bcd and △qrs congruent?

Explanation:

Step1: Find the side lengths of \(\triangle BCD\)

Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) or counting units on the grid.
For \(BC\): Counting units, \(BC = 9\) (horizontal units).
For \(CD\): Using the distance formula between \(C(0,10)\) and \(D(-9,0)\), \(CD=\sqrt{(0 + 9)^2+(10 - 0)^2}=\sqrt{81 + 100}=\sqrt{181}\).
For \(BD\): Counting units, \(BD = 10\) (vertical units).

Step2: Find the side lengths of \(\triangle QRS\)

For \(QR\): Counting units, \(QR = 9\) (horizontal units).
For \(RS\): Using the distance formula between \(R(0,- 10)\) and \(S(9,0)\), \(RS=\sqrt{(9 - 0)^2+(0 + 10)^2}=\sqrt{81+100}=\sqrt{181}\).
For \(QS\): Counting units, \(QS = 10\) (vertical units).

Step3: Check for congruence

Since \(BC = QR = 9\), \(CD=RS=\sqrt{181}\), \(BD = QS = 10\), by SSS (Side - Side - Side) congruence criterion.

Answer:

\(\triangle BCD\) and \(\triangle QRS\) are congruent.