Question 4

Find the area of triangle ADE in Fig. 6.31.

The parallel sides of a trapezium are $40\text{ cm}$ and $20\text{ cm}$. If its non-parallel sides are both equal, each being $26\text{ cm}$, find the area of the trapezium.

Find the area of a triangle, given that its sides are $8\text{ cm}$ and $11\text{ cm}$ long, and its perimeter is $32\text{ cm}$.

One diagonal of a rhombus is twice as long as the other diagonal. If the rhombus has area $128\text{ cm}^2$, find the length of the shorter diagonal.

$ABCD$ is a parallelogram. $P$ and $Q$ are any two points on side $AB$. What can you say about the ratio $\text{area}(\Delta \text{PCD}) : \text{area}(\Delta \text{QCD})$?

$O$ is any point on the diagonal $PR$ of a parallelogram $PQRS$. Prove that the areas of triangles $PSO$ and $PQO$ are equal.

If the mid-points of the sides of a 4-gon (also known as a quadrilateral, but we prefer to call it a ‘4-gon’) are joined in order, prove that the area of the parallelogram thus formed will be half of the area of the given 4-gon. (You may wonder whether the 4-gon thus formed is always a parallelogram, and if so, why? These questions will be tackled and answered in the chapter on quadrilaterals.)

In $\Delta\text{ABC}$, the midpoint of $\text{BC}$ is $\text{D}$ (Fig. 6.32). Median $\text{AD}$ is drawn. $\text{P}$ is any point on $\text{AD}$. Show that $\text{area}(\Delta\text{ABP}) = \text{area}(\Delta\text{ACP})$.

Given a square $ABCD$, let $P$ be a point within it. Join $PA$, $PB$, $PC$, $PD$ (Fig. 6.33). What is the ratio of the areas of the red region ($\Delta PAB$ and $\Delta PCD$) and the green region ($\Delta PBC$ and $\Delta PDA$)?

In $\Delta\text{ABC}$, $D$ is the midpoint of $\text{AB}$. $P$ is any point on $\text{BC}$, and $Q$ is a point on $\text{AB}$ such that $\text{CQ} \parallel \text{PD}$. $\text{PQ}$ is joined (Fig. 6.34). Prove that

$\text{Area}(\Delta\text{BPQ}) = \dfrac{1}{2} \text{Area}(\Delta\text{ABC})$