(a) The diagonals of a rhombus are 10.2 cm…

Question 1 Report

(a) The diagonals of a rhombus are 10.2 cm and 9.3 cm long. Calculate, correct to one decimal place, the perimeter of the rhombus.

(b) Given that \(\sin x = \frac{3}{5}, 0° < x < 90°\), find the value of \(5\cos x - 4\tan x\).

Answer

Answer Details

(a) The diagonals of a rhombus bisect each other at right angles. Half-diagonals are

\[ \frac{10.2}{2} = 5.1\ \text{cm}, \qquad \frac{9.3}{2} = 4.65\ \text{cm}. \]

A side of the rhombus is the hypotenuse of a right triangle with these legs:

\[ s = \sqrt{5.1^2 + 4.65^2} = \sqrt{26.01 + 21.6225} = \sqrt{47.6325} = 6.9016\ \text{cm}. \]

Perimeter \(= 4s = 4 \times 6.9016 = 27.6\ \text{cm (to 1 d.p.).}\)

(b) Given \(\sin x = \tfrac{3}{5}\) with \(0^\circ < x < 90^\circ\), this is a 3-4-5 right triangle, so

\[ \cos x = \frac{4}{5}, \qquad \tan x = \frac{3}{4}. \]

Then

\[ 5\cos x - 4\tan x = 5\left(\frac{4}{5}\right) - 4\left(\frac{3}{4}\right) = 4 - 3 = 1. \]

(a) The diagonals of a rhombus are 10.2 cm and 9.3 cm long. Calculate, correct to one decimal place, the perimeter of the rhombus. (b) Given that \(\sin x =...