Question

In the given figure express `bar"c"` and `bar"d"` in terms of `bar"a"` and `bar"b"`.

Answer 1

`bar"PQ" = bar"PS" + bar"SQ"`

∴ `bar"a" = bar"c" - bar"d"`   ...(1)

`bar"PR" = bar"PS" + bar"SR"`

∴ `bar"b" = bar"c" + bar"d"`    ....(2)

Adding equations (1) and (2), we get,

`bar"a" + bar"b" = (bar"c" - bar"d") + (bar"c" + bar"d") = 2bar"c"`

∴ `bar"c" = 1/2 (bar"a" + bar"b")`

∴ `bar"c" = 1/2 bar"a" + 1/2bar"b"`

Substituting for `bar"c"` in (2), we get,

∴ `bar"b" = bar"c" + bar"d"`

∴ `bar"d" = bar"b" − bar"c"`

∴ `bar"d" = bar"b" − 1/2 (bar"a" + bar"b")`

∴ `bar"d" = (2bar"b" − (bar"a" + bar"b"))/2`

∴ `bar"d" = (2bar"b" − bar"a" − bar"b")/2`

∴ `bar"d" = (bar"b" − bar"a")/2`

∴ `bar"d" = 1/2 (bar"b" − bar"a")`

∴ `bar"d" = 1/2bar"b" - 1/2bar"a"`

Hence, `bar"c" = 1/2 bar"a" + 1/2bar"b"` and `bar"d" = 1/2bar"b" - 1/2bar"a"`.