Question

!50 POINTS! (3 SIMPLE GEOMETRY QUESTIONS)

QUESTIONS BELOW
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Answer 1

First = b. 32.5 ft

Second = d. 7√2 cm

Third = a. 45.7*

Explanation:

First question

Sin 38* = 20 ft / total length of 6 cars

total length of 6 cars = 20 ft / 0.61566147532

= 32.5 ft

Second question

given that,

diagonal of a square = 14 cm

diagonal of a square = a√2

so,

14 cm = a√2 cm

a = 14/√2

= 14√2/√2√2

= 14√2/2

= 7√2 cm

Third question

tan x = 8/7.8

tan x = 1.02564102564

x = 45.7*

Answer 2

10) b) 32.5 ft

11) d) 7√2 cm

12) a) 45.7°

Explanation:

Question 10

The given scenario can be modelled as a right triangle, where the base of the triangle is 20 ft, the angle between the height of the triangle and the hypotenuse is 38°.

To find the length of the six cars, we need to find the length of the hypotenuse. To do this we can use the sine trigonometric ratio.

`\boxed{\begin{minipage}{9 cm}\underline{Sine trigonometric ratio} \\\\$\sf \sin(\theta)=(O)/(H)$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}`

Given values:

• θ = 38°
• O = 20 ft
• H = H (to be found)

Substitute the given values into the sine ratio and solve for H:

`\begin{aligned}\sin 38^(\circ)&=(20)/(H)\\\\H}&=(20)/(\sin 38^(\circ))\\\\H}&=32.4853849...\\\\H}&=32.5\; \sf ft\;(nearest\;tenth)\end{aligned}`

Therefore, the total length of the six cars is approximately 32.5 ft (rounded to the nearest tenth).

`\hrulefill`

Question 11

In a square, the diagonals form right triangles, where the diagonal is the hypotenuse, and the sides of the square are the legs. As the sides of a square are congruent, the legs of the right triangle are equal in length. Therefore, the triangle is a special 45-45-90 right triangle.

The special property of a 45-45-90 triangle is that the hypotenuse is equal to the length of either leg multiplied by √2. This means that the length of the leg is the length of the hypotenuse divided by √2.

Given the diagonal of the square is 14 cm, then the hypotenuse of the right triangle is 14 cm. Therefore, the length of the leg (x) can be calculated as:

`\begin{aligned}x&=(14)/(√(2))\\\\x&=(14 \cdot √(2))/(√(2)\cdot √(2))\\\\x&=(14 √(2))/(2)\\\\x&=7 √(2)\end{aligned}`

Therefore, the length of the legs of the right triangle are 7√2 cm, so the side length of the square is 7√2 cm.

`\hrulefill`

Question 12

The given right triangle has legs measuring 7.8 units and 8 units.

To find the measure of angle x (the angle opposite the leg measuring 8 units), we can use the tangent trigonometric ratio.

`\boxed{\begin{minipage}{7 cm}\underline{Tangent trigonometric ratio} \\\\$\sf \tan(\theta)=(O)/(A)$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle.\\\end{minipage}}`

Given values:

• θ = x
• O = 8
• A = 7.8

Substitute the given values into the tangent ratio and solve for x°:

`\begin{aligned}\tan x^(\circ)&=(8)/(7.8)\\\\x^(\circ)&=\tan^(-1)\left((8)/(7.8)\right)\\\\x^(\circ)&=45.725224...^(\circ)\\\\x^(\circ)&=45.7^(\circ)\; \sf (nearest\;tenth)\end{aligned}`

Therefore, the measure of angle x is 45.7° (rounded to the nearest tenth).