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## 'Two Ladders' printed from http://nrich.maths.org/

The picture shows two ladders propped against facing walls. The end of the first ladder is $5\text{m}$ above the base of the second wall. The second ladder is $5\text{m}$ above the base of the first wall.

**Can you work out at what height the ladders cross?**

Are you surprised that we don't need to know the distance between the two walls?

Suppose now the first ladder is $10 \text{m}$ above the base of the second wall, with the second ladder remaining as before.

**Can you work out at what height the ladders cross now?**

Click below to reveal a diagram that may help you get started:

Can you find any pairs of similar triangles in this diagram?

How do these help you to find the value of $x$?

Suppose the first ladder meets the wall at a height of $a$ metres, as in the diagram below.

**At what height do the ladders cross now?**

Suppose the second ladder meets the wall at a height of $b$ metres, as in the diagram below.

**At what height do the ladders cross now?**