Question

In the given figure, m(arc WY) = 44°, m(arc ZX) = 68°, then
(1) Find the measure of ∠ ZTX.
(2) If WT = 4.8, TX = 8.0,
YT = 6.4, find TZ.
(3) If WX = 25, YT = 8,
YZ = 26, find WT.

Answer 1

XW and YZ are two chords of a circle intersecting each other in the interior of the circle at T.

(1)
If two chords of a circle intersect each other in the interior of a circle then the measure of the angle between them is half the sum of measures of arcs intercepted by the angle and its opposite angle.

∴ ∠ZTX = $\frac{1}{2}$[m(arc ZX) + m(arc WY)] = $\frac{1}{2}\times \left(68°+44°\right)=\frac{1}{2}\times 112°$ = 56º

Thus, the measure of ∠ZTX is 56º.

(2)
WT × TX = YT × TZ (Theorem of internal division of chords)

⇒ 4.8 × 8 = 6.4 × TZ

⇒ TZ = $\frac{4.8\times 8}{6.4}$ = 6

(3)
WT × TX = YT × TZ (Theorem of internal division of chords)

⇒ WT × (WX − WT) = YT × (YZ − YT)

⇒ WT × (25 − WT) = 8 × (26 − 8)

⇒ 25WT − WT² = 8 × 18 = 144

⇒ WT² − 25WT + 144 = 0

⇒ WT² − 16WT − 9WT + 144 = 0

⇒ WT(WT − 16) − 9(WT − 16) = 0

⇒ (WT − 16)(WT − 9) = 0

⇒ WT − 16 = 0 or WT − 9 = 0

⇒ WT = 16 or WT = 9