Question

1. find the perimeter of the triangle: image of coordinate grid with points a(-2, 3), c(-2, -3), b(3, -3) options: a. √61 − 11, b. √61 + 11, c. √61, d. 11
2. find the area of the triangle: image of coordinate grid with point a(-2, 3)

Explanation:

Question 10:

Step1: Find length of AC

Points A(-2, 3) and C(-2, -3). Since x - coordinates are same, distance is difference in y - coordinates.
$AC=\vert3 - (-3)\vert=\vert6\vert = 6$

Step2: Find length of BC

Points B(3, -3) and C(-2, -3). Since y - coordinates are same, distance is difference in x - coordinates.
$BC=\vert3-(-2)\vert=\vert5\vert = 5$

Step3: Find length of AB

Use distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ for A(-2, 3) and B(3, -3).
$AB=\sqrt{(3 - (-2))^2+(-3 - 3)^2}=\sqrt{(5)^2+(-6)^2}=\sqrt{25 + 36}=\sqrt{61}$

Step4: Find perimeter

Perimeter = AC+BC + AB=6 + 5+\sqrt{61}=\sqrt{61}+11

The triangle is a right - triangle with base BC = 5 and height AC = 6 (or base AC = 6 and height BC = 5). The formula for the area of a right - triangle is $A=\frac{1}{2}\times base\times height$.

Step1: Identify base and height

Base (BC) = 5, Height (AC)=6

Step2: Calculate area

$A=\frac{1}{2}\times5\times6 = 15$

(If the triangle in question 11 has the same coordinates as question 10, the area is 15. But since the diagram for question 11 is partially shown, if we assume the vertical side from A(-2,3) to some point (say ( - 2,y)) and the slant side, but from the first triangle, if it's the same right - triangle, area is 15. If the second triangle in question 11 has points A(-2,3), let's say the other two points are ( - 2, y) and (x, y) (from the partial diagram), but with the given first triangle's pattern, if we take the vertical segment from A(-2,3) down (length, say, let's assume the vertical side is length h and horizontal side length b, but from the first triangle, if it's a right - triangle with legs of length, for example, if the vertical leg is from y = 3 to y = - 1 (assuming the lower point is ( - 2,-1)) and horizontal leg from x=-2 to x = 0 (but this is speculative). However, from the first triangle (question 10), the area is 15. If the second triangle is similar or same, but since the diagram is partial, based on the first triangle:

For triangle with vertices A(-2,3), C(-2,-3), B(3,-3) (same as question 10), it's a right triangle.

Step1: Determine base and height

Base $BC = 5$ (horizontal distance), Height $AC=6$ (vertical distance)

Step2: Apply area formula

Area $=\frac{1}{2}\times base\times height=\frac{1}{2}\times5\times6 = 15$

Answer:

B. $\sqrt{61}+11$

Question 11:

(Assuming the triangle is the same as in question 10 with points A(-2,3), B(3,-3), C(-2,-3))