Question

find the perimeter of the triangle. (round to the nearest tenth if needed)

if b is the midpoint of \\( \overline { a c } \\), find the value of m.
10.

11.
find the area of the circle with a diameter of 15 inches.

Explanation:

Step1: Find the lengths of the sides of the triangle

Assume the vertices of the right - triangle are \(A(2,1)\), \(B(2,8)\), \(C(7,1)\).
The length of \(AB\) (vertical side): Using the distance formula \(d=\vert y_2 - y_1\vert\), for \(A(2,1)\) and \(B(2,8)\), \(AB=\vert8 - 1\vert=7\).
The length of \(BC\) (horizontal side): Using the distance formula \(d=\vert x_2 - x_1\vert\), for \(B(2,8)\) and \(C(7,1)\), \(BC=\vert7 - 2\vert = 5\).
The length of \(AC\) (hypotenuse): Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 5\) and \(b = 7\), \(AC=\sqrt{5^{2}+7^{2}}=\sqrt{25 + 49}=\sqrt{74}\approx8.6\).

Step2: Calculate the perimeter of the triangle

The perimeter \(P\) of a triangle is \(P=a + b + c\). Here \(a = 5\), \(b = 7\), \(c\approx8.6\). So \(P=5 + 7+8.6=20.6\).

Step1: Use the mid - point property

If \(B\) is the mid - point of \(\overline{AC}\), then \(AB = BC\). Given \(AB = 3m+5\) and \(BC = 4m - 10\).
Set up the equation \(3m+5=4m - 10\).

Step2: Solve the equation for \(m\)

Subtract \(3m\) from both sides: \(3m+5-3m=4m - 10-3m\), which gives \(5=m - 10\).
Add 10 to both sides: \(m=5 + 10=15\).

Step1: Find the radius of the circle

The formula for the area of a circle is \(A=\pi r^{2}\), and the radius \(r=\frac{d}{2}\). Given \(d = 15\) inches, then \(r=\frac{15}{2}=7.5\) inches.

Step2: Calculate the area of the circle

Substitute \(r = 7.5\) into the area formula \(A=\pi r^{2}\), \(A=\pi\times(7.5)^{2}=\pi\times56.25 = 56.25\pi\) square inches.

Answer:

B. 20.6