Question

rectangle abcd is on a coordinate plane with vertices a(1,4), b(4,1), c(8,5), and d(5,8).
triangle pqr is also on a coordinate plane with vertices p(8,1), q(12,3), and r(9,4), with a right angle at ∠r.
find the exact areas of rectangle abcd and △pqr using the distance formula. do not use decimal approximations.
use the keypad to enter the answers in the boxes.
the area of rectangle abcd is square units.
the area of △pqr is square units.

Explanation:

Step1: Calculate the length of \(AB\)

The distance formula is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(A(1,4)\) and \(B(4,1)\), \(AB=\sqrt{(4 - 1)^2+(1 - 4)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\).

Step2: Calculate the length of \(BC\)

For \(B(4,1)\) and \(C(8,5)\), \(BC=\sqrt{(8 - 4)^2+(5 - 1)^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}\).

Step3: Calculate the area of rectangle \(ABCD\)

The area of a rectangle \(A = l\times w\). Here \(l = AB\) and \(w = BC\), so \(A_{ABCD}=3\sqrt{2}\times4\sqrt{2}=24\).

Step4: Calculate the length of \(PR\)

For \(P(8,1)\) and \(R(9,4)\), \(PR=\sqrt{(9 - 8)^2+(4 - 1)^2}=\sqrt{1 + 9}=\sqrt{10}\).

Step5: Calculate the length of \(QR\)

For \(Q(12,3)\) and \(R(9,4)\), \(QR=\sqrt{(9 - 12)^2+(4 - 3)^2}=\sqrt{9+1}=\sqrt{10}\).

Step6: Calculate the area of \(\triangle PQR\)

The area of a right - triangle \(A=\frac{1}{2}a\times b\) (where \(a\) and \(b\) are the legs). Here \(a = PR\) and \(b = QR\), so \(A_{\triangle PQR}=\frac{1}{2}\times\sqrt{10}\times\sqrt{10}=5\).

Answer:

The area of rectangle \(ABCD\) is \(24\) square units.
The area of \(\triangle PQR\) is \(5\) square units.