Question

Rhombus QRST has diagonals intersecting at W. Point U is located on side QR and point V on diagonal RT such that UV is perpendicular to QR. Prove: QW*UR = WT*UV

Answer 1

Answer:

just see on Google and on Socratic app

Answer 2

Rhombus QRST has diagonals intersecting at W. Point U is located on side QR and point V on diagonal RT such that UV is perpendicular to QR.

Consider the attached figure while going through the following steps.

Given:

A rhombus QRST with diagonals intersecting at W.

Point U is located on side QR and point V on diagonal RT.

UV is perpendicular to QR

To prove:

QW * UR = WT * UV

Proof:

∠ QWR = ∠ QWT = ∠ RWS = ∠ TWS = 90°

(interior angles of rhombus are 90°)

QW≅WS and WR ≅ WT

(diagonals of rhombus  bisect perpendicularly)

In Δ QWR and Δ UVR,

∠ WRQ = ∠ VRU    (common angles)

∠ QWR = ∠ VUR     (rt. angles)

 ∴ Δ QWR ~ Δ UVR      (using AA criteria)

⇒ QW / VU = WR / UR     (c.p.c.t)

QW × UR = WR × UV

QW × UR = WT × UV               (∵ WR ≅ WT )

Hence proved.