11.0: Triangles and Circles

1.1 Angles and Triangles

Homework 1.1

1.

3.

5.

7. $\theta = 108.8^{\circ}$

9. $\alpha = 29^{\circ}$

11. $\beta = 77^{\circ}$

13. $\alpha = 12^{\circ}$

15. $\theta = 65^{\circ}$

17. $\theta = 22^{\circ}$

19. $\psi = 73^{\circ}$

21. $\phi = 88^{\circ}$

23.

a $\phi = 120^{\circ}$

b $\phi=160^{\circ}$

c $\phi=\alpha+\beta$

d An exterior angle is equal to the sum of the opposite interior angles.

25. $\theta=72^{\circ}, \phi=54^{\circ}$

27. $\theta=100^{\circ}, \phi=30^{\circ}$

29.

a $180^{\circ}$

b $90^{\circ}$

c a right triangle

31.

a $180^{\circ}$

b $90^{\circ}$

c a right triangle

33. $\alpha=30^{\circ}, \beta=60^{\circ}$

35. $x=47^{\circ}, y=133^{\circ}$

37. $x=60^{\circ}, y=15^{\circ}$

39. $x=25^{\circ}, y=16^{\circ}$

41. $x=90^{\circ}, y=55^{\circ}$

43. $x=50^{\circ}, y=80^{\circ}$

45.

a $\angle 1=\angle 4, \angle 3=\angle 5$

b $180^{\circ}$

c In the equation $\angle 4+\angle 2+\angle 5=180^{\circ}$, substitute $\angle 1$ for $\angle 4$, and substitute $\angle 3$ for $\angle 5$ to conclude that the sum of the angles in the triangle is $180^{\circ}$.

47. $\angle 1=130^{\circ}$ because vertical angles are equal. $\angle 2=50^{\circ}$ because it makes a straight angle with a $130^{\circ}$ angle. $\angle 3=65^{\circ}$ because it is a base angle of an isosceles triangle whose vertex angle is $50^{\circ}$. $\angle 41=65^{\circ}$ for the same reason. $\angle 5=25^{\circ}$ because it is complementary to $\angle 4$.

1.2 Similar Triangles

Homework 1.2

1. $\triangle P Q T \simeq \triangle S R T, x=7, y=3, \alpha=18^{\circ}$

3. $\triangle P R E \simeq \triangle U R N, z=12, \theta=10^{\circ}, \phi=70^{\circ}$

5. $\triangle ABT \simeq \triangle ABC, \text{ so } AT = AC$

7. Similar. Corresponding sides are proportional.

9. Similar. Corresponding angles are equal.

11. $\angle A=37^{\circ}, \angle B=37^{\circ}$

13. $h=12$

15. $p = 35$

17. $g = 84$

19. $h = 30$

21. 154 feet

23. 1 mile

25. 17.1 square feet

27. $y=\dfrac{12}{17} x$

29. $h = 7.5$

31. $c = 15$

33. $s = 6$

35. $y=\dfrac{3}{5} x$

37. $y=5+\dfrac{3}{4} x$

39.

a $\angle B=70^{\circ}, \angle C A D=70^{\circ}, \angle D A B=20^{\circ}$

b $\triangle D B A$ and $\triangle D B C$. The hypotenuse is $B C$ in $\triangle A B C, B A$ in $\triangle D B A$, and $A C$ in $\triangle D A C$. The short leg is $A B$ in $\triangle A B C, D B$ in $\triangle D B A$, and $D A$ in $\triangle D A C$. The longer leg is $A C$ in $\triangle A B C, D A$ in $\triangle D B A$, and $D C$ in $\triangle D A C$.

1.3 Circles

Homework 1.3

1. 13 miles

3. 10, 10.0

5. $4 \sqrt{5} \approx 8.94$

7. 5

9. $2 \sqrt{5}$

13.

24.7

15.

a $\sqrt{(x+3)^2+(y-4)^2}$

b $\sqrt{(x+3)^2+(y-4)^2}=5$

17. The distance between the points $(x, y)$ and (4,-1) is 3 units.

19.

a $6 \sqrt{2} \mathrm{~cm}$

b $8.49 \mathrm{~cm}$

21.

a $25 \pi \mathrm{sq}$ in

b $78.54 \mathrm{sq}$ in

23.

a approximation

b approximation

c approximation

d exact

25.

a

$x$	-5	-4	-3	-2	-1	0	1	2	3	4	5
$y$	0	$\pm 3$	$\pm 4$	$\pm \sqrt{21}$	$\pm 2\sqrt{6}$	$\pm 5$	$\pm 2 \sqrt{6}$	$\pm \sqrt{21}$	$\pm 4$	$\pm 3$	0

b

27.

a

b $x^2+y^2 = 36$

29.

a

b $x^2+y^2 < 9$

31.

a No real value of $y$ can satisfy $x^2+y^2 = 16$ unless $-4 \leq x \leq 4$

b The graph has no points where $x > 4$ and no points where $x < -4$

33. $\sqrt{10}$

35.

a

b $12 \pi$

37.

a

b $4\pi$

39. $(-2\sqrt{5}, -4), (2\sqrt{5}, -4)$

41. $P\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right), Q\left(\dfrac{1}{2}, \dfrac{-\sqrt{3}}{2}\right), R\left(\dfrac{-3}{4}, \frac{\sqrt{7}}{4}\right), S\left(\dfrac{-3}{4}, \dfrac{-\sqrt{7}}{4}\right)$

43.

a $45^{\circ}$

b $5 \pi \mathrm{ft}$

c $50 \pi \mathrm{sq} \mathrm{ft}$

45.

a $\dfrac{2}{5}$

b $40 \pi \mathrm{sq} \mathrm{ft}$

c $8 \pi \mathrm{ft}$

46.

a $\dfrac{1}{10}$

b $\dfrac{\pi}{10} \mathrm{sq} \mathrm{km}$

c $\dfrac{\pi}{5} \mathrm{~km}$

47.

a $\dfrac{5}{6}$

b $\dfrac{15 \pi}{2}$ sq m

c $5 \pi \mathrm{m}$

51. 2070 miles

53.

a 54,000 miles

b 2240 mph

55.

a $(x-3)^2+(y+2)^2=36$

b $(x-h)^2+(y-k)^2=r^2$

1.4 Chapter 1 Summary and Review

Chapter 1 Review Problems

1.

3.

5. $\alpha = \beta = \gamma = 60^{\circ}$

7. $\phi = \omega = 79^{\circ}$

9. $\theta = 65^{\circ}, \phi = 25^{\circ}$

11. $\delta = 30^{\circ}, \gamma = 60^{\circ}$

13. $\sigma = 39^{\circ}, \omega = 79^{\circ}$

15. $\alpha = 51 \dfrac{3}{7}^{\circ}, \beta = 64 \dfrac{2}{7}^{\circ}$

17. $\triangle A B C \cong \triangle E D C, \alpha=40^{\circ}, \beta=130^{\circ}, x=32$

19. Yes, three pairs of equal angles

21. Yes, three pairs of equal angles

23. 13

25. 18

27. $y = \dfrac{5x}{2}$

29. $y=\dfrac{7x}{3}$

31. $y = \dfrac{x}{3}$

33. $x = \dfrac{25}{13}, y = \dfrac{60}{13}$

35. $\alpha = 70^{\circ}$

37. 14 ft

39. $3\dfrac{3}{4}$in

41. All side have length $\sqrt{61}$, opposite sides have slopes $\frac{5}{6}$ and $\frac{-6}{5}$

43. $AC = BC = 18$

45.

a $\sqrt{(x-2)^2+(y-5)^2}=3$

b $(x-2)^2+(y-5)^2=9$

47. $4 \sqrt{5} \approx 8.944 \mathrm{~cm}$

49. $\left(\dfrac{-1}{3}, \dfrac{2 \sqrt{2}}{3}\right),\left(\dfrac{-1}{3}, \dfrac{-2 \sqrt{2}}{3}\right)$

51.

a $4 \pi \mathrm{ft}$

b $20 \pi \mathrm{ft}^2$

53.

a $45^{\circ}, 60^{\circ}$

b $\dfrac{49 \pi}{8}$ in $^2, 6 \pi$ in $^2$ Delbert

c $\dfrac{79 \pi}{4}$ in, $2 \pi$ in, Francine