Question

Please help me! it's my third time asking

Answer 1

The beach, the parking lot and the refreshment stand are illustrations of similar triangles.

• The distance between the spot and the parking lot is 24 m

• The distance between the refreshment stand and the parking lot is 40 m

(a) The distance between the spot and the parking lot

Represent this distance with d.

So, the equivalent ratio is:

`\small \bold{ 32: d = d: 18 }`

Express as fractions

`\small \bold{(32)/(d) =(d)/(18)}`

Cross multiply

`\small \bold{ d × d = 32 * 18 }`

`\small \bold{d² = 576 }`

Take square roots

`\small \bold{d = 24 }`

Hence, the distance between the spot and the parking lot is 24 m

(b) The distance between the refreshment stand and the parking lot

Represent this distance with d.

Using Pythagoras theorem we have:

`\small \bold{ d² = 32² +24² }`

`\small \bold{d² = 1600 }`

Take square roots

`\small \bold{ d = 40}`

Hence, the distance between the refreshment stand and the parking lot is 40 m

Answer 2

Beach to parking lot = 24 m

Parking lot to refreshment stand = 40 m

Explanation:

Part a

Right triangle altitude theorem:

`(a)/(h)=(h)/(b)`

This theorem describes the relationship between the altitude (h) on the hypotenuse in a right triangle and the two line segments (a and b) it creates on the hypotenuse. (a is the shorter segment and b is the longer segment of the hypotenuse).

For the given triangle:

• h = the distance between the parking lot and the beach
• a = 18 m
• b = 32 m

Substituting these values into the formula:

`(18)/(h)=(h)/(32)`

`\implies 18 \cdot 32=h^2`

`\implies h^2=576`

`\implies h=√(576)`

`\implies h=24 \textsf{ m}`

So the distance between the beach and the parking lot is 24 m

Part b

We can now use Pythagoras' Theorem to calculate the distance between the parking lot and the refreshment stand.

Pythagoras' Theorem: a² + b² = c²

(where a and b are the legs, and c is the hypotenuse, of a right triangle)

Given:

• a = 24
• b = 32
• c = distance between refreshment stand and parking lot

Substituting these values into the formula:

⇒ 24² + 32² = c²

⇒ c² = 1600

⇒ c = √1600

⇒ c = 40 m

Therefore, the distance between the parking lot and the refreshment stand is 40 m.