Across the River

Part of the Triangles section of Coddy's Geometry journey. Lesson 47 of 60.

Surveyors measure a river they cannot cross with a pair of similar triangles. Stand at A, straight opposite a tree T on the far bank. Walk along the bank to a stake B, then on to C. From C walk straight away from the river until you see B and T in a line: that is D.

$A tree across a river with stakes on the near bank making two similar triangles$

The two triangles TAB and DCB have right angles at A and C, and the angles at B are vertical angles, so they are equal. By AA they are similar, with A matching C. Here AB = 15, BC = 5 and CD = 6, so $\frac{w}{6} = \frac{15}{5}$ and the river is 18 m wide.

Challenge

A tree T stands across a river, straight opposite A. The stakes give AB = 20 m and BC = 8 m along the bank, and CD = 12 m straight back from the river, with D, B and T in a line. Find the width AT, then press Check.

Try it yourself

$Find the marked part of the figure$