Question

1. find perimeter of parallelogram abcd. round to the nearest tenth.
2. answer the following:

a. what is the length of the base of the triangle shown?
b. what is the height of the triangle shown?
c. what is the formula for an area of a triangle?
d. calculate the area of the triangle.

1. name the following polygons. are they convex or concave?
2. what is the m∠xyz? is the angle acute, right or obtuse?

Explanation:

Step1: Find side - lengths of parallelogram

Assume coordinates of vertices of parallelogram \(ABCD\) are \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\), \(D(x_4,y_4)\). Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the lengths of two adjacent sides, say \(AB\) and \(AD\). Since opposite sides of a parallelogram are equal, perimeter \(P=2(AB + AD)\).
Let's assume \(A(- 5,-2)\), \(B(5,-2)\), \(D(-3,2)\) (by observing the grid).
\(AB=\sqrt{(5 + 5)^2+(-2+2)^2}=\sqrt{10^2}=10\)
\(AD=\sqrt{(-3 + 5)^2+(2 + 2)^2}=\sqrt{2^2+4^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\approx4.5\)

Step2: Calculate perimeter

\(P = 2(AB+AD)=2(10 + 4.5)=2\times14.5 = 29.0\)

Step3: For triangle in question 9

a. By observing the grid of the triangle, count the number of units along the base - the base length \(b\). Assume the base is horizontal, and by counting grid - squares, \(b = 8\) units.
b. Count the number of units from the base to the opposite vertex to get the height \(h\). The height \(h=6\) units.
c. The formula for the area of a triangle is \(A=\frac{1}{2}bh\).
d. Substitute \(b = 8\) and \(h = 6\) into the formula: \(A=\frac{1}{2}\times8\times6=24\) square units.

Step4: For question 10

a. The first polygon is a hexagon. Since all interior angles are less than \(180^{\circ}\), it is convex.
b. The second polygon is a star - shaped polygon (a non - regular octagon in a sense). It has interior angles greater than \(180^{\circ}\), so it is concave.
c. The third polygon is a pentagon. Since all interior angles are less than \(180^{\circ}\), it is convex.

Step5: For question 11

By reading the protractor, the measure of \(\angle XYZ\) is \(110^{\circ}\). Since \(90^{\circ}<110^{\circ}<180^{\circ}\), the angle is obtuse.

Answer:

1. Perimeter of parallelogram \(ABCD\) is \(29.0\)
2. a. Base length of the triangle is \(8\)

b. Height of the triangle is \(6\)
c. \(A=\frac{1}{2}bh\)
d. Area of the triangle is \(24\) square units

1. a. Hexagon, convex

b. Star - shaped polygon, concave
c. Pentagon, convex

1. \(m\angle XYZ = 110^{\circ}\), obtuse