Question

In the figure below, PDW and WTC are right triangles. The measure of WPD is 30°, the measure of WC is 6 units, the measure of WT is 3 units, and the measure of WD is 12 units. Determine the measure of PD .

Answer 1

Explanation:

Using similarity it can be solved as:

`In\:\triangle PDW\: \& \: \triangle CTW\\\\</p><p>\angle PDW \cong \angle CTW.... (each\:90 °) \\\\</p><p>\angle PWD \cong \angle CWT... (Vertical\:\angle s) \\\\</p><p>\therefore \triangle PDW\: \sim\: \triangle CTW\\</p><p>(by \:AA\: criterion \: of\: similarity) \\\\</p><p>\therefore (PW)/(CW) =(WD)/(WT).. (csst) \\\\</p><p></p><p>\therefore (PW)/(6) =(12)/(3)\\\\</p><p></p><p>\therefore (PW)/(6) =4\\\\</p><p>\therefore PW=4* 6\\\\</p><p>\huge \red {\boxed {\therefore PW=24}} \\\\</p><p>In\:\triangle PDW\:, \angle PWD = 60°\\\\</p><p>\therefore PD =\frac {\sqrt 3}{2} * PW\\\\</p><p></p><p>\therefore PD =\frac {\sqrt 3}{2} * 24\\\\</p><p>\huge \orange {\boxed {\therefore PD = 12\sqrt 3\: units}}`

Answer 2

PD = 12
`√(3)`

Explanation:

Using the tangent ratio in right Δ PDW and the exact value

tan30° =
`(1)/(√(3) )`, then

tan30° =
`(opposite)/(adjacent)` =
`(WD)/(PD)` =
`(12)/(PD)` =
`(1)/(√(3) )` ( cross- multiply )

PD = 12
`√(3)`