Across the River

Part of the Triangles section of Coddy's Geometry journey. Lesson 58 of 71.

You can measure the width of a river without crossing it. Stand at A, directly opposite a tree T on the far bank. Walk 15 m along the bank to B and plant a stake there. Walk 5 m further along the bank to C. Then turn away from the river and walk until the stake lines up with the tree. In the figure this takes 9 m, to the point D. The sight line from T to D passes through the stake.

$A river drawn as two dashed lines; a tree T on the far bank directly across from A; along the near bank A to B is 15 and B to C is 5; from C a line of 9 leads away from the river to D, and the sight line from T through the stake at B reaches D$

Two right triangles meet at the stake

The two triangles TAB and DCB are similar. Triangle TAB has a right angle at A, and triangle DCB has a right angle at C. The angles at the stake B are equal, because two straight lines cross there. So the river width x matches the 9, and the 15 matches the 5. Write the proportion and solve it:

x/9 = 15/5

x = 27

The river is 27 m wide, and every measurement was taken on the near bank.

Challenge

Point A is directly opposite a tree across the river. The walk along the bank from A to the stake at B is 20 m, and from B to C is a further 8 m. From C, walking 12 m away from the river to D lines the stake up with the tree.

$A river drawn as two dashed lines; a tree T on the far bank directly across from A; along the near bank A to B is 20 and B to C is 8; from C a line of 12 leads away from the river to D, and the sight line from T through the stake at B reaches D$

Find the width x of the river.

Try it yourself

x/12 = 20/8

Solve for x

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

Across the River

All lessons in Triangles