Question

practice busing formulas in geometryuse the figures for exercises 1-3.1. find the perimeter of triangle a.2. find the area of triangle a.3. triangle a is identical to triangle b.find the height $h$ of triangle b.find the perimeter and area of each shape.4. square with a side 2.4 m in length5. rectangle with length $(x + 3)$ and width 76. although a circle does not have sides, it does have a perimeter.what is the term for the perimeter of a circle?find the circumference and area of each circle.7. use $\frac{22}{7}$ for $pi$.8. use 3.14 for $pi$.9. leave $pi$ as $pi$.10. the area of a square is $\frac{1}{4}$ $\text{in}^2$. find the perimeter.11. the area of a triangle is $152$ $\text{m}^2$, and the height is 16 m. find the base.12. the circumference of a circle is $25pi$ mm. find the radius.use the figure for exercises 13 and 14.lucas has a 39-foot-long rope. he uses all the rope to outline this t-shape in his backyard. all the angles in the figure are right angles.13. find $x$.14. find the area enclosed by the rope.

Explanation:

Step1: Sum all sides of Triangle A

$3 + 4 + 5 = 12$

Step2: Apply triangle area formula for A

$\text{Area} = \frac{1}{2} \times 3 \times 4 = 6$

Step3: Set equal areas for A and B

$\frac{1}{2} \times 5 \times h = 6$
Solve for $h$: $h = \frac{12}{5} = 2.4$

Step4: Square perimeter and area

Perimeter: $4 \times 2.4 = 9.6$
Area: $2.4^2 = 5.76$

Step5: Rectangle perimeter and area

Perimeter: $2[(x+3)+7] = 2x + 20$
Area: $7(x+3) = 7x + 21$

Step6: Identify circle perimeter term

Recall standard geometry terminology

Step7: Circle 7 circumference and area

Radius $r = \frac{14}{2} = 7$
Circumference: $2 \times \frac{22}{7} \times 7 = 44$
Area: $\frac{22}{7} \times 7^2 = 154$

Step8: Circle 8 circumference and area

Circumference: $2 \times 3.14 \times 1.5 = 9.42$
Area: $3.14 \times 1.5^2 = 7.065$

Step9: Circle 9 circumference and area

Circumference: $2\pi(x+1) = 2\pi x + 2\pi$
Area: $\pi(x+1)^2 = \pi(x^2 + 2x + 1)$

Step10: Find square side then perimeter

Side length $s = \sqrt{\frac{1}{4}} = \frac{1}{2}$
Perimeter: $4 \times \frac{1}{2} = 2$

Step11: Solve for triangle base

Rearrange area formula: $b = \frac{2 \times 152}{16} = 19$

Step12: Solve for circle radius

Rearrange circumference formula: $r = \frac{25\pi}{2\pi} = \frac{25}{2} = 12.5$

Step13: Set perimeter equal to rope length

$2(9 + 3 + x) + 2(9 - 3.5 - 3.5) = 39$
Simplify: $2(12 + x) + 2(2) = 39$
$24 + 2x + 4 = 39$
$2x = 11$
$x = 5.5$

Step14: Calculate T-shape area

Area = Area of top rectangle + Area of bottom rectangle
$= 9 \times 3 + (9 - 3.5 - 3.5) \times 5.5$
$= 27 + 2 \times 5.5 = 27 + 11 = 38$

Answer:

1. $12$ ft
2. $6$ ft$^2$
3. $2.4$ ft
4. Perimeter: $9.6$ m, Area: $5.76$ m$^2$
5. Perimeter: $2x + 20$, Area: $7x + 21$
6. Circumference
7. Circumference: $44$ mi, Area: $154$ mi$^2$
8. Circumference: $9.42$ cm, Area: $7.065$ cm$^2$
9. Circumference: $2\pi x + 2\pi$, Area: $\pi x^2 + 2\pi x + \pi$
10. $2$ in
11. $19$ m
12. $12.5$ mm
13. $5.5$ ft
14. $38$ ft$^2$