1. One side of a parallelogram is \( \color{black} 14cm \). Its distance from the opposite side is \( \color{black} 16.5cm \). Find the area of the parallelogram.
Solution
\[ \begin{align*} \text{Base of parallelogram }(AB) &= 14cm \\ \text{Height }(DE) &= 16cm \\ \\ \text{Area of the parallelogram} &= \text{Base} \times \text{Height} \\ &= 14cm \times 16.5cm \\ &= \color{green} 231cm^2 \\ \end{align*} \]
Answer Area of the parallelogram \( = \color{red} 231cm^2 \)
2. A parallelogram has a base of \( \color{black} 135 \, dm \). The corresponding height is \( \color{black} 6 \, dm \). Find the area of the parallelogram in square metres.
Solution
\[ \begin{align*} \text{Base of parallelogram }(AB) &= 135dm \\ \text{Height }(DE) &= 6dm \\ \\ \text{Convert to } & metre \\ 1 \, dm &= \frac{1}{10}m \\ \\ \text{Base} &= 135 \, dm \\ & = \frac{135}{10} \\ & = 13.5m \\ \\ \text{Height} &= 6 \, dm \\ & = \frac{6}{10} \\ & = 0.6m \\ \\ \text{Area of the parallelogram} &= \text{Base} \times \text{Height} \\ &= 13.5m \times 0.6m \\ &= \color{green} 8.1m^2 \\ \end{align*} \]
Answer Area of the parallelogram \( = \color{red} 8.1m^2 \)
3. The altitude of a parallelogram corresponding to a base of length \( \color{black} 15cm \) is \( \color{black} 18cm \). Find the area of the parallelogram.
Solution
\[ \begin{align*} \text{Base of parallelogram }(AB) &= 15cm \\ \text{Height }(DE) &= 18cm \\ \\ \text{Area of the parallelogram} &= \text{Base} \times \text{Height} \\ &= 15cm \times 18cm \\ &= \color{green} 270cm^2 \\ \end{align*} \]
Answer Area of the parallelogram \( = \color{red} 270cm^2 \)
4. The height of a parallelogram is \( \color{black} 3 \, dm \). If the area is \( \color{black} 240 \, cm^2 \), find the base of the parallelogram.
Solution
\[ \begin{align*} \text{Height }(DE) &= 3 \, dm \\ \text{Convert height from } & dm \text{ to } cm \\ 1 \, dm &= 10 \, cm \\ 3 \, dm &= 3 \times 10 \\ &= 30 \, cm \\ \\ \text{Area of the parallelogram} &= 240 \, {cm}^2 \\ Base \times Height & = 240cm^2 \\Base \times 30 & = 240 \\ \\ Base &= \frac{240}{30} \\ \\ Base &= \color{green} 8cm \end{align*} \]
Answer Base of the parallelogram \( = \color{red} 8cm \)
5. Find the area of a rhombus whose one side is \( \color{black} 8cm \) and altitude is \( \color{black} 0.6dm \).
Solution
\[ \begin{align*} \text{Side of rhombus }(AB) &= 8cm \\ \text{Altitude }(DE) &= 0.6dm \\ 1 \, dm & = 10 \, cm \\ 0.6dm &= 0.6 \times 10 \\ &= 6cm \\ \\ \text{Area of the rhombus} &= \text{Side} \times \text{Altitude} \\ &= 8cm \times 6cm \\ &= \color{green} 48cm^2 \\ \end{align*} \]
Answer Area of the rhombus \( = \color{red} 48cm^2 \)
6. Find the altitude of a rhombus whose area is \( \color{black} 320 m^2 \) and side is \( \color{black} 5m \).
Solution
\[ \begin{align*} \text{Area of rhombus} &= 320m^2 \\ \text{Side of rhombus }(AB) &= 5m \\ \\\text{Altitude} &= \frac{Area}{Side} \\ \\ &= \frac{320m^2}{5m} \\ \\ \text{Altitude} &= \color{green} 64m \\ \end{align*} \]
Answer Altitude of the rhombus \( = \color{red} 64m \)
7. The height of a parallelogram is one-third of its base. If the area is \( \color{black} 108 \, cm^2 \), find the base and height.
Solution
\[ \begin{align*} \text{Let base of parallelogram } &= x \, \text{ cm} \\ \text{Let height of parallelogram } &= \frac{x}{3} \, \text{ cm} \\ \\ \text{Area of the parallelogram} &= 108 \, cm^2 \\ Base \times Height & = 108cm^2 \\ \\x \times \frac{x}{3} & = 108 \\ \\ \frac{x^2}{3} & = 108 \\ x^2 & = 108 \times 3 \\ x^2 & = 324 \\ x^2 & = 18 \times 18 \\ x^2 & = {18}^2 \\ x & = \sqrt{{18}^2} \\ x & = 18cm \\ \\ Base &= x \implies 18cm \\ \\ Height &= \frac{x}{3} \\ \\ &= \frac{18}{3} \\ \\ Height &= 6cm \\\end{align*} \]
Answer Base of the parallelogram \( = \color{red} 18 \, cm \), Height of the parallelogram \( = \color{red} 6 \, cm \)
8. The area of a rhombus is \( \color{black} 119 \, cm^2 \) and its perimeter is \( \color{black} 56 \, cm \). Find its altitude.
Solution
\[ \begin{align*} \text{Perimeter of rhombus} &= 56 \, cm \\ 4 \times \text{Side} &= 56 \, cm \\ \\ Side &= \frac{56 \, cm}{4} \\ \\ Side&= 14 \, cm \\ \\ \text{Area of rhombus} &= 119 \, cm^2 \\ \\ Side \times Altitude &= 119 \\ 14 \times Altitude &= 119 \\ \\ Altitude &= \frac{\cancelto{17}{119}}{\cancelto{2}{14}} \\ \\ &= \frac{17}{2} \\ \\ Altitude &= \color{green} 8.5 \, cm \\ \end{align*} \]
Answer Altitude of the rhombus \( = \color{red} 8.5 \, cm \)
9. One side and corresponding altitude of a parallelogram are \( \color{black} 50 \, cm \) and \( \color{black} 8 \, cm \). If the other altitude is \( \color{black} 4 \, cm \), find the length of other pair of parallel sides.
Solution
\[ \begin{align*} \text{Area of parallelogram (ABCD)} &= \text{Area of parallelogram (ABCD)} \\ \text{Base} \times \text{Height} &= \text{Base} \times \text{Height} \\ \text{AB} \times \text{DE} &= \text{DA} \times \text{BF} \\ 50 \times 8 &= \text{DA} \times 4 \\ \\ \frac{50 \times \cancelto{2}{8}}{\cancelto{1}{4}} &= \text{DA} \\ \\ 100 \, cm &= \text{DA} \\ \\ \end{align*} \]
Answer The length of the other pair of parallel sides \( = \color{red} 100 \, cm \)
10. Area of a parallelogram is \( \color{black} 625 \, m^2 \). Find the length of sides of parallelogram if altitudes corresponding to sides are \( \color{black} 20 \, m \) and \( \color{black} 25 \, m \).
Solution
\[ \begin{align*} \text{Altitude } (DE) &= 20 \, m \\ \text{Area of parallelogram} &= 625 m^2 \\ Base \times Altitude &= 625 m^2 \\ AB \times 20 &= 625 m^2 \\ \\ AB &= \frac{\cancelto{125}{625}}{\cancelto{4}{20}} \\ \\ &= \frac{125}{5} \\ \\ AB &= \color{green} 31.25 \, m \\ \\\text{Altitude } (BF) &= 25 \, m \\ \text{Area of parallelogram} &= 625 m^2 \\ Base \times Altitude &= 625 m^2 \\ AD \times 25 &= 625 m^2 \\ \\ AD &= \frac{\cancelto{125}{625}}{\cancelto{5}{25}} \\ \\ &= \frac{125}{5} \\ \\ AD &= \color{green} 25 m \\ \end{align*} \]
Answer Length of sides of parallelogram \( = \color{red} 31.25 m \) and \( = \color{red} 25 m \)