Question

1. the vertices of a triangle are at (1, 3), (1, 9), and (-7, 3). find the perimeter of the triangle

Explanation:

Step1: Calculate the length of the vertical side

The two points \((1,3)\) and \((1,9)\) have the same \(x -\)coordinate. Using the distance formula \(d=\vert y_2 - y_1\vert\), we have \(d_1=\vert9 - 3\vert=6\).

Step2: Calculate the length of the horizontal side

The two points \((1,3)\) and \((-7,3)\) have the same \(y -\)coordinate. Using the distance formula \(d=\vert x_2 - x_1\vert\), we have \(d_2=\vert-7 - 1\vert = 8\).

Step3: Calculate the length of the hypotenuse

Use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 6\) and \(b = 8\). So \(c=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\).

Step4: Calculate the perimeter

The perimeter \(P\) of a triangle is \(P=d_1 + d_2 + c\). Substitute \(d_1 = 6\), \(d_2=8\) and \(c = 10\) into the formula. Then \(P=6 + 8+10=24\).

Answer:

\(24\)