Question

1. calculate the distance of each side. round your answers to the nearest hundredth place.

side 1: endpoints are (1,1) and (4,3)
d = \sqrt{(4 - 1)^2 + (3 - 1)^2}
d = \sqrt{3^2 + 2^2}
d \approx 3.61

1. does the completed figure on the coordinate grid have right angles? explain and justify.

1. is the completed figure on the coordinate grid a square? explain and justify.

Explanation:

Question 5 Solution (assuming we need to re - calculate or verify the distance)

Step 1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 1,y_1 = 1,x_2=4,y_2 = 3\) (assuming the second endpoint is \((4,3)\) as there might be a typo in the original \((41,3)\)).

Step 2: Substitute values into the formula

Substitute \(x_1 = 1,y_1 = 1,x_2 = 4,y_2=3\) into the formula: \(d=\sqrt{(4 - 1)^2+(3 - 1)^2}=\sqrt{3^2+2^2}\)

Step 3: Calculate the values inside the square root

First, calculate \(3^2=9\) and \(2^2 = 4\). Then \(3^2+2^2=9 + 4=13\)

Step 4: Take the square root and round

\(d=\sqrt{13}\approx3.61\) (since \(\sqrt{13}\approx3.6055\) and rounding to the nearest hundredth gives \(3.61\))

Answer:

The distance of the side with endpoints \((1,1)\) and \((4,3)\) is approximately \(3.61\)

Question 6 Brief Explanations:

To determine if the figure has right angles, we can use the slope formula (\(m=\frac{y_2 - y_1}{x_2 - x_1}\)) to find the slopes of the sides. If the product of the slopes of two adjacent sides is \(- 1\), then the sides are perpendicular (forming a right angle). For example, if we have two sides with slopes \(m_1\) and \(m_2\), and \(m_1\times m_2=-1\), the angle between them is \(90^{\circ}\). We can also use the Pythagorean theorem (if we know the lengths of the sides). If for a triangle formed by three vertices (a corner of the figure), the sum of the squares of the lengths of two shorter sides equals the square of the length of the longest side (\(a^{2}+b^{2}=c^{2}\)), then the triangle is a right triangle, and the figure has a right angle at that vertex.