Question

42 which ordered pair represents a solution to the equation... (image of a coordinate plane with a line) a (-2, 2) b (6, 0) c (4, 1) d (2, -3) 43 the vertices of triangle ghj are g(8, 0), h(6, -4), and j(0, -1). what kind of triangle is ghj? a right scalene b non - right scalene c right isosceles d non - right isosceles

Explanation:

Question 42

Step1: Recall solution of linear equation

A solution to the equation of a line is an ordered pair \((x,y)\) that lies on the line. So we check which of the given ordered pairs is on the graphed line.

Step2: Check each option

• Option A: \((-2, 2)\) – Plotting \(x = -2\), \(y = 2\) on the grid, this point is not on the line.
• Option B: \((6, 0)\) – Plot \(x = 6\), \(y = 0\). From the graph, the line crosses the x - axis at \(x = 6\) (since when \(y = 0\), \(x = 6\)), so this point is on the line.
• Option C: \((4, 1)\) – Plot \(x = 4\), \(y = 1\). This point is not on the line.
• Option D: \((2, - 3)\) – Plot \(x = 2\), \(y=-3\). This point is not on the line.

Step1: Calculate the lengths of the sides

To determine the type of triangle, we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\).

• Length of \(GH\): For \(G(8,0)\) and \(H(6, - 4)\)

\(GH=\sqrt{(6 - 8)^2+(-4 - 0)^2}=\sqrt{(-2)^2+(-4)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)

• Length of \(HJ\): For \(H(6, - 4)\) and \(J(0, - 1)\)

\(HJ=\sqrt{(0 - 6)^2+(-1+4)^2}=\sqrt{(-6)^2+(3)^2}=\sqrt{36 + 9}=\sqrt{45}=3\sqrt{5}\)

• Length of \(GJ\): For \(G(8,0)\) and \(J(0, - 1)\)

\(GJ=\sqrt{(0 - 8)^2+(-1 - 0)^2}=\sqrt{(-8)^2+(-1)^2}=\sqrt{64 + 1}=\sqrt{65}\)

Step2: Check for right triangle (Pythagorean theorem)

We check if \(GH^{2}+HJ^{2}=GJ^{2}\) (or other combinations).
\(GH^{2}=(2\sqrt{5})^{2}=20\), \(HJ^{2}=(3\sqrt{5})^{2}=45\), \(GJ^{2}=65\)
Since \(20 + 45=65\), \(GH^{2}+HJ^{2}=GJ^{2}\), so the triangle is right - angled.

Step3: Check for isosceles or scalene

Since \(GH = 2\sqrt{5}\), \(HJ = 3\sqrt{5}\), \(GJ=\sqrt{65}\), all sides have different lengths (because \(2\sqrt{5}
eq3\sqrt{5}
eq\sqrt{65}\)), so it is scalene.

Answer:

B. \((6, 0)\)

Question 43