Question

road construction!

the city of interlaken has three amazing natural recreational areas: a lake, a cliff, and a mountainous area. currently, there are two businesses open in the area, half moon kayak rentals and chupacabra campground. two new adventure companies want to open their businesses in this area to attract adventure-seeking tourists. however, the issue is the only road that runs through this area is lombard road. the city wants to build more roads to make the recreational areas accessible to tourists.

construct the roads with a ruler and protractor to meet the citys criteria (2 points)

interlakens road criteria

• a straight road must begin at chupacabra campground and end at ijump adventure company.
• a straight road must begin at half moon kayak rentals and end at surge bike excursions.
• a straight road must pass by ijump adventure co. and surge bike excursions but never intersect lombard road.
• name the roads and provide appropriate angle measures of all intersecting roads for construction.

Explanation:

Analyze the road criteria

We need to construct three straight roads connecting four locations:

1. Chupacabra Campground (middle-left, near Lombard Road)
2. Half Moon Kayak Rentals (upper-right, near the lake)
3. iJump Adventure Company (bottom-left, near the cliff)
4. Surge Bike Excursions (bottom-right, near the mountains)

The criteria are:

• Road 1: Connects Chupacabra Campground and iJump Adventure Company.
• Road 2: Connects Half Moon Kayak Rentals and Surge Bike Excursions.
• Road 3: Passes through iJump Adventure Co. and Surge Bike Excursions, but never intersects Lombard Road.
• Name the roads and provide appropriate angle measures of all intersecting roads.

Identify parallel relationships

Using the Parallel Line Angles and Corresponding Angles Postulate knowledge points

• Since Road 3 (connecting iJump and Surge) must never intersect Lombard Road, Road 3 must be parallel to Lombard Road.
• Let Lombard Road be line \(L_1\) and Road 3 be line \(L_2\), where \(L_1 \parallel L_2\).
• Road 1 (Chupacabra to iJump) and Road 2 (Half Moon to Surge) act as transversals intersecting these parallel lines.

Determine angle relationships

Using the Alternate Interior Angles knowledge point

• Let the transversal Road 1 intersect Lombard Road at Chupacabra Campground with an angle \(\theta_1\), and intersect Road 3 at iJump with an angle \(\phi_1\). Since the roads are parallel, the alternate interior angles are equal: \(\theta_1 = \phi_1\).
• Let the transversal Road 2 intersect Lombard Road at Half Moon Kayak Rentals with an angle \(\theta_2\), and intersect Road 3 at Surge with an angle \(\phi_2\). Similarly, the alternate interior angles are equal: \(\theta_2 = \phi_2\).

Define road names and design values

Let's assign clear names and realistic geometric values for construction:

• Lombard Road: The existing baseline road.
• Adventure Way (Road 3): The new road connecting iJump and Surge. It is parallel to Lombard Road.
• Campground Trail (Road 1): Connects Chupacabra and iJump.
• Kayak Pass (Road 2): Connects Half Moon and Surge.

Let's define a set of consistent angle measures for construction:

• Angle between Lombard Road and Campground Trail: \(75^\circ\)
• Angle between Adventure Way and Campground Trail: \(75^\circ\) (alternate interior angle)
• Angle between Lombard Road and Kayak Pass: \(60^\circ\)
• Angle between Adventure Way and Kayak Pass: \(60^\circ\) (alternate interior angle)

Answer:

To meet the city's criteria, construct the roads as follows:

1. Road 1: Campground Trail

• Route: Connects Chupacabra Campground to iJump Adventure Company.
• Intersection Angle: Intersects Lombard Road at an angle of \(75^\circ\).

1. Road 2: Kayak Pass

• Route: Connects Half Moon Kayak Rentals to Surge Bike Excursions.
• Intersection Angle: Intersects Lombard Road at an angle of \(60^\circ\).

1. Road 3: Adventure Way

• Route: Connects iJump Adventure Company to Surge Bike Excursions.
• Alignment: Constructed parallel to Lombard Road so they never intersect.
• Intersection Angles:
• Intersects Campground Trail (Road 1) at an angle of \(75^\circ\) (alternate interior angle to the Lombard Road intersection).
• Intersects Kayak Pass (Road 2) at an angle of \(60^\circ\) (alternate interior angle to the Lombard Road intersection).