Source: various maths texts! Strange Guys Source: various maths texts!

Strange Guys 900 𝑚 100 𝑚 Not to scale ℎ 𝑚 350 𝑚 Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross?

Strange Guys ℎ 𝑥 = 𝑎 𝑑 and ℎ 𝑑−𝑥 = 𝑏 𝑑 𝑥= ℎ𝑑 𝑎 and 𝑑−𝑥= ℎ𝑑 𝑏 From similar triangles: ℎ 𝑥 = 𝑎 𝑑 and ℎ 𝑑−𝑥 = 𝑏 𝑑 Rearranging: 𝑥= ℎ𝑑 𝑎 and 𝑑−𝑥= ℎ𝑑 𝑏 Adding gives: 𝑑= ℎ𝑑 𝑎 + ℎ𝑑 𝑏

Strange Guys 𝑑=ℎ𝑑 1 𝑎 + 1 𝑏 1=ℎ 𝑎+𝑏 𝑎𝑏 So ℎ= 𝑎𝑏 𝑎+𝑏 Or 1 ℎ = 1 𝑎 + 1 𝑏 𝑥 𝑑 𝑑= ℎ𝑑 𝑎 + ℎ𝑑 𝑏 𝑑=ℎ𝑑 1 𝑎 + 1 𝑏 1=ℎ 𝑎+𝑏 𝑎𝑏 So ℎ= 𝑎𝑏 𝑎+𝑏 Or 1 ℎ = 1 𝑎 + 1 𝑏 Notice that the distance between the masts is irrelevant to the question asked.

The guys cross 90 𝑚 above the ground. Strange Guys 𝑏 𝑎 ℎ 𝑑 For: 𝑎=100 𝑏=900 ℎ= (100) 900 100+900 =90 The guys cross 90 𝑚 above the ground.

Strange Guys Can you simply explain why the distance 𝑑 is irrelevant? ℎ 𝑏 𝑎 𝑑 Can you simply explain why the distance 𝑑 is irrelevant? Imagine the diagram was printed on a rubber sheet and then stretched parallel to the ground.

Strange Guys Can you simply explain why the distance 𝑑 is irrelevant? 𝑏 𝑎 Can you simply explain why the distance 𝑑 is irrelevant? Imagine the diagram was printed on a rubber sheet and then stretched parallel to the ground. You can see that ℎ is unaltered by the transformation. Note also that the ratio 𝑥:𝑦 is unaltered.

RESOURCES

Strange Guys 900 𝑚 100 𝑚 Not to scale ℎ 𝑚 350 𝑚 Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? SIC_35

Strange Guys Strange Guys ℎ ℎ Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 108 𝑚 540 𝑚 350 𝑚 ℎ 𝑚 Not to scale SIC_35 Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 110 𝑚 495 𝑚 360 𝑚 ℎ 𝑚 Not to scale SIC_35

Strange Guys Strange Guys ℎ ℎ Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 115 𝑚 414 𝑚 550 𝑚 ℎ 𝑚 Not to scale SIC_35 Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 117 𝑚 390 𝑚 580 𝑚 ℎ 𝑚 Not to scale SIC_35

Strange Guys Strange Guys ℎ ℎ Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 120 𝑚 360 𝑚 455 𝑚 ℎ 𝑚 Not to scale SIC_35 Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 126 𝑚 315 𝑚 215 𝑚 ℎ 𝑚 Not to scale SIC_35

Strange Guys Strange Guys ℎ ℎ Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 135 𝑚 270 𝑚 256 𝑚 ℎ 𝑚 Not to scale SIC_35 Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 140 𝑚 252 𝑚 512 𝑚 ℎ 𝑚 Not to scale SIC_35

Strange Guys Strange Guys ℎ ℎ Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 144 𝑚 240 𝑚 452 𝑚 ℎ 𝑚 Not to scale SIC_35 Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 150 𝑚 225 𝑚 555 𝑚 ℎ 𝑚 Not to scale SIC_35

Strange Guys Strange Guys ℎ ℎ Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 165 𝑚 198 𝑚 200 𝑚 ℎ 𝑚 Not to scale SIC_35 Two tall vertical masts on horizontal, level ground have guy ropes tightly tethered to the other’s base. They cross at a point between them. How high above the ground do they cross? Strange Guys 171 𝑚 190 𝑚 533 𝑚 ℎ 𝑚 Not to scale SIC_35