Gilligan sees a ship coming close to the shore he's standing on. He wants to determine the distance (segment SD) from the ship to the shore. He walks 130 ft along the shore from point D to point C and marks that spot. Then he walks 23 ft further and marks point B. He turns 90°and walks until his location (point A), point C, and point S are collinear.

Answer the following questions making sure to show all your work.

(a) Can Gilligan conclude that triangle ABC and triangle SDC are similar? Why or why not?

(b) Suppose AB = 150 ft. What is the distance from the ship to the shore? Show all your work and round your final answer to the nearest tenth of a foot.

The AA similarity theorem states that, if two angles in one triangle are congruent to two corresponding angles in another triangle, then both triangles are similar triangles. The corresponding sides of both triangles would be proportional.

a. Triangles ABC and SDC have two pairs of congruent angles, which are:

∠SCD ≅ ∠ACB (vertical angles are congruent)

∠SDC ≅ ∠ABC (right angles = 90°)

Therefore, Gilligan can conclude that ΔABC ~ ΔSDC by AA similarity theorem.

b. Given, AB = 150 ft

BC = 23 ft

DC = 130 ft

Distance from ship to the shore = SD = ?

Thus:

AB/SD = BC/DC (proportional sides of similar triangles).

Substitute

150/SD = 23/130

SD = (130 × 150)/23

SD = 847.8 ft.

Therefore, distance from the ship to the shore, to the nearest tenth of a foot, is: 847.8.

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