Solve the geometry problems as shown in the image.

Steps to Solve

Question 8

1. Solve for $x$ In a rhombus, the diagonals bisect the angles. Therefore, $m\angle 1 = m\angle 2$.

$2x + 20 = 5x - 4$

$24 = 3x$

$x = 8$

2. Solve for $AF$ Since diagonals of a rhombus bisect each other, $AF = \frac{1}{2}AC$. Given that $AC = 15$, then $AF = \frac{1}{2}(15) = 7.5$.

3. Find $m\angle 3$ The diagonals of a rhombus are perpendicular, creating right angles where they intersect. Therefore, $m\angle AFC = 90^{\circ}$. The angles in the triangle $AFC$ add to $180^{\circ}$, so $m\angle 1 + m\angle 3 + m\angle AFC = 180$ and $m\angle 1 + m\angle 3 = 90$.

$m\angle 1 = 2x + 20 = 2(8) + 20 = 16 + 20 = 36$

$m\angle 3 = 90 - 36 = 54$

4. Solve for $y$ We have the equation for $m\angle 3 = y^2 + 26$.

$y^2 + 26 = 54$

$y^2 = 28$

$y = \sqrt{28} = 2\sqrt{7}$

Question 9

1. Find the side length A square has 4 equal sides, so its perimeter is $P = 4s$, where $s$ is the side length. Given $P = 162$, $s = \frac{162}{4} = 40.5$ meters.

2. Find the area The area of a square is $A = s^2$. $A = (40.5)^2 = 1640.25$ square meters

Question 10

1. Divide quadrilateral into triangles Quadrilateral $ABCD$ can be divided into two triangles, triangle $ADC$ and triangle $ABC$. We can find the area of each triangle and sum the areas.

2. Find the area of triangle $ADC$ The base, $DC$, is $4$, and the height is $3$. The area, $A_1$, of $ADC$ can be obtained:

$A_1 = \frac{1}{2} \cdot 4 \cdot 3 = 6$

3. Find the area of triangle $ABC$ The base, $AC$, is $7$, and the height of $ABC$ is $1.5$

Area, $A_2$, can be obtained:

$A_2 = \frac{1}{2} \cdot 7 \cdot 1.5 = 5.25$

4. The area of $ABCD$ $A = A_1 + A_2 = 6 + 5.25 = 11.25$ square inches.

Question 11

1. Find $AB$ The diagonals of a rhombus bisect each other at right angles. Thus, we have a right triangle with legs $\frac{AC}{2}$ and $\frac{BD}{2}$. $\frac{AC}{2} = \frac{12}{2} = 6$ and $\frac{BD}{2} = \frac{16}{2} = 8$. $AB$ is the hypotenuse of this triangle.

$AB = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm.

2. Find the perimeter of $ABCD$ Since a rhombus has four equal sides, the perimeter is $4 \times AB = 4 \times 10 = 40$ cm.

3. Find the area of $ABCD$ Area of a rhombus is $\frac{1}{2} \times d_1 \times d_2$ where $d_1$ and $d_2$ are the lengths of the diagonals. $A = \frac{1}{2} \times 12 \times 16 = 96$ square cm.