Question

1. (5-11) review: geometry copy the trapezoid at right on your paper. then find its area and perimeter. keep your work organized so that you can later explain how you solved it. (note: the diagram is not drawn to scale.) 5. (5-12) review: pre - algebra solve each of the equations below for the given variable. be sure to check your answers. a. 4(2x + 5) - 11 = 4x - 3 b. $\frac{2m - 1}{19}=\frac{m}{10}$

Explanation:

4.

Step1: Find the length of the non - parallel side

Since the right - triangle formed on the left of the trapezoid has an angle of 45° and height 4', and it is a 45 - 45-90 triangle, the base of this right - triangle is also 4' (because in a 45 - 45-90 triangle, the legs are equal). Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = b=4\), we get \(c=\sqrt{4^{2}+4^{2}}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\) feet.

Step2: Calculate the area of the trapezoid

The formula for the area of a trapezoid is \(A=\frac{(b_1 + b_2)h}{2}\), where \(b_1 = 15\), \(b_2=22\), and \(h = 4\). So \(A=\frac{(15 + 22)\times4}{2}=\frac{37\times4}{2}=74\) square feet.

Step3: Calculate the perimeter of the trapezoid

The perimeter \(P\) of the trapezoid is the sum of all its sides. The sides are 15, 4\sqrt{2}, 22, and the other non - parallel side (which is also 4\sqrt{2} by symmetry). So \(P=15+4\sqrt{2}+22 + 4\sqrt{2}=37 + 8\sqrt{2}\approx37+8\times1.414=37 + 11.312 = 48.312\) feet.

Step1: Expand the left - hand side

Expand \(4(2x + 5)-11\) using the distributive property. \(4(2x+5)-11=8x + 20-11=8x + 9\). So the equation becomes \(8x + 9=4x-3\).

Step2: Move the \(x\) terms to one side

Subtract \(4x\) from both sides: \(8x-4x+9=4x-4x - 3\), which simplifies to \(4x+9=-3\).

Step3: Move the constant to the other side

Subtract 9 from both sides: \(4x+9 - 9=-3 - 9\), so \(4x=-12\).

Step4: Solve for \(x\)

Divide both sides by 4: \(x=\frac{-12}{4}=-3\).

Step1: Cross - multiply

Cross - multiply the equation \(\frac{2m - 1}{19}=\frac{m}{10}\) to get \(10(2m - 1)=19m\).

Step2: Expand the left - hand side

Expand \(10(2m - 1)\) using the distributive property: \(20m-10 = 19m\).

Step3: Move the \(m\) terms to one side

Subtract \(19m\) from both sides: \(20m-19m-10=19m-19m\), so \(m - 10=0\).

Step4: Solve for \(m\)

Add 10 to both sides: \(m=10\).

Answer:

Area: 74 square feet; Perimeter: \(37 + 8\sqrt{2}\approx48.312\) feet

5.

a.