Question

which statement about △arm is true? the length of (overline{rm}) is 4 units. the length of (overline{ar}) is (sqrt{84}) units. the area is 15 square units. the perimeter is 22 units.

Explanation:

Step1: Find coordinates of points

Assume \(A=(0,5)\), \(R = (- 5,-4)\), \(M=(0,-5)\)

Step2: Calculate length of \(RM\)

Using distance formula for two - points \((x_1,y_1)\) and \((x_2,y_2)\) \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(R(-5,-4)\) and \(M(0,-5)\), \(x_1=-5,y_1 = - 4,x_2=0,y_2=-5\). Then \(RM=\sqrt{(0 + 5)^2+(-5 + 4)^2}=\sqrt{25 + 1}=\sqrt{26}
eq4\).

Step3: Calculate length of \(AR\)

For \(A(0,5)\) and \(R(-5,-4)\), \(x_1 = 0,y_1=5,x_2=-5,y_2=-4\). \(AR=\sqrt{(-5 - 0)^2+(-4 - 5)^2}=\sqrt{25+81}=\sqrt{106}
eq\sqrt{84}\).

Step4: Calculate area of \(\triangle ARM\)

Base \(AM\): \(AM=\vert5-(-5)\vert = 10\), height (horizontal distance from \(R\) to \(y - axis\)) \(h = 5\). Area \(A=\frac{1}{2}\times base\times height=\frac{1}{2}\times10\times3 = 15\) square units.

Step5: Calculate perimeter

We already know \(AM = 10\), \(AR=\sqrt{106}\), \(RM=\sqrt{26}\). Perimeter \(P=10+\sqrt{106}+\sqrt{26}
eq22\).

Answer:

The area is 15 square units.