In the diagram. PQR is an isosceles triangle. If the perimeter of the triangle is 28 cm, find the: a. values of x and y; b. lengths of the sides of the tria...

Answer Details

In triangle PQR the three sides are marked as follows: \(|PQ| = 2y + x\), \(|QR| = 4y\) and \(|PR| = 6y - 2x + 1\). Since PQR is isosceles the two slant sides are equal, \(|QR| = |PR|\).

(a) Values of x and y

Perimeter equation:

\[ (2y + x) + 4y + (6y - 2x + 1) = 28 \]

\[ 12y - x + 1 = 28 \quad\Rightarrow\quad 12y - x = 27 \quad\text{...(1)} \]

Isosceles equation \((|QR| = |PR|)\):

\[ 4y = 6y - 2x + 1 \quad\Rightarrow\quad 2x - 2y = 1 \quad\text{...(2)} \]

From (1), \(x = 12y - 27\). Substituting into (2):

\[ 2(12y - 27) - 2y = 1 \Rightarrow 24y - 54 - 2y = 1 \Rightarrow 22y = 55 \]

\[ y = \frac{55}{22} = \frac{5}{2} = 2\tfrac{1}{2} \]

\[ x = 12\left(\tfrac{5}{2}\right) - 27 = 30 - 27 = 3 \]

Hence x = 3 and y = 2\(\tfrac{1}{2}\) (i.e. 2.5).

(b) Lengths of the sides

• \(|PQ| = 2y + x = 2\left(\tfrac{5}{2}\right) + 3 = 5 + 3 = \textbf{8 cm}\)
• \(|QR| = 4y = 4\left(\tfrac{5}{2}\right) = \textbf{10 cm}\)
• \(|PR| = 6y - 2x + 1 = 6\left(\tfrac{5}{2}\right) - 2(3) + 1 = 15 - 6 + 1 = \textbf{10 cm}\)

Check: \(8 + 10 + 10 = 28\) cm, which agrees with the given perimeter, and \(|QR| = |PR| = 10\) cm confirms the triangle is isosceles.