DAV Class 7 Maths Chapter 8 Worksheet 3
Triangle And Its Properties Worksheet 3
1. Which of the following can be the sides of a triangle? Measurement of sides are given in centimetres.
(a) 6, 4, 8
Solution:
\begin{align*} \text{Sum of two sides} && \quad \text{Third Side} \\ \end{align*} \begin{align*} 6 + 4 &= 10 & > && 8 \\ 4 + 8 &= 12 & > && 6 \\ 8 + 6 &= 14 & > && 4 \\ \end{align*}
\({\boxed{ \color{red} \text{Triangle is possible}}}\), because sum of the lengths of any two sides of a triangle is always greater than the length of third side.(b) 8, 10, 18
Solution:
\begin{align*} \text{Sum of two sides} && \quad \text{Third Side} \\ \end{align*} \begin{align*} 8 + 10 &= 18 & = && 18 \\ 10 + 18 &= 28 & > && 8 \\ 8 + 18 &= 26 & > && 10 \\ \end{align*}
\({\boxed{\color{red} \text{Triangle is not possible}}}\).(c) 3, 4, 10
Solution:
\begin{align*} \text{Sum of two sides} && \quad \text{Third Side} \\ \end{align*} \begin{align*} 3 + 4 &= 7 & < && 10 \\ 4 + 10 &= 14 & > && 3 \\ 3 + 10 &= 13 & > && 4 \\ \end{align*}
\({\boxed{ \color{red} \text{Triangle is not possible}}}\).(d) 35, 38, 40
Solution:
\begin{align*} \text{Sum of two sides} && \quad \text{Third Side} \\ \end{align*} \begin{align*} 35 + 38 &= 73 & > && 40 \\ 38 + 40 &= 78 & > && 35 \\ 35 + 40 &= 75 & > && 38 \\ \end{align*}
\({\boxed{\text{Triangle is possible}}}\), because sum of the lengths of any two sides of a triangle is always greater than the length of third side.2. A point O is in the interior of ▵ABC. Use the symbols >, < or = to make the following statements true.
(a) OA + OB _____ AB
Solution: OA + OB \(\color{red} {\Large\boxed{ > }}\) AB
(b) OB + OC _____ BC
Solution: OB + OC \( \color{red} {\Large\boxed{ > }}\) BC
(c) OB + OC _____ BC
Solution: OB + OC \( \color{red} {\Large\boxed{ > }}\) BC
Show that AB + BC + AC < 2 ( OA + OB+ OC )
Solution:
\begin{align*} \text{We know that} \\ OA + OB & > AB \\ OB + OC & > BC \\ OA + OC & > AC \\ \\ \text{Adding all the 3 conditions we get} \\ \\ OA + OB +OB + OC + OA + OC & > \text{AB + AB + BC} \\ 2 \text{OA} + 2 \text{OB} + 2 \text{OC} & > \text{AB + AB + BC} \\ 2 \text{(OA + OB + OC)} & > \text{AB + AB + BC} \\ \\ \implies AB + AB + BC & < 2 (OA + OB + OC) \end{align*}
(3) The length of two sides of a triangle are 4 cm and 6 cm. Between what two measures should the length of the third side fall.
Solution:
\begin{align*} &\text{Given two sides of a triangle: } 4\text{ cm and } 6\text{ cm.} \\ &\text{Let the the third side be } (x) \\ \\ 1.& \quad 4 + 6 > x \implies x < 10\\ 2.& \quad x + 6 > 4 \implies x > -2 \text{ (Not possible, lengths are positive)} \\ 3.& \quad 4 + x > 6 \implies x > 2 \\ \\ &\text{Therefore, the third side must be more than 2 and less than 10} & \\ & \text{ The measure of third side must lie between 3 cm and 9 cm }. \end{align*}