Across the Lake

Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 1 of 44.

A surveyor wants the distance to a tree T on the far side of a lake. She sets two posts A and B 150 m apart on her shore and measures the angle at each post between the baseline and the line to the tree: 48° at A, 64° at B. The angles of a triangle add up to 180°, so the angle at the tree is 68°.

$Posts A and B 150 m apart, the tree T across the lake, angles 48°, 64° and 68°$

There is no right angle, so SOH CAH TOA has nothing to work on. The law of sines does: in any triangle, each side divided by the sine of the angle across from it gives the same number.

a/sin(A) = b/sin(B)

The distance d from A to the tree is across from the 64° angle, and the baseline is across from 68°, so d/sin(64°) = 150/sin(68°). Multiply both sides by sin(64°): d = 150sin(64°)/sin(68°), about 145.41 m.

On the board, tap an angle, the side across from it and a second such pair, then pick Law of sines from the menu. The board writes the equation from the figure; you solve it and round.

Challenge

Two posts A and B stand 120 m apart on the shore. The angle between the baseline and the line to the tree T is 52° at A and 61° at B, so the angle at the tree is 67°. Find the distance x from A to the tree, to two decimal places. Tap the parts you need, pick the rule, solve, and press Check.

Try it yourself

$Use the figure to find x$