Class 9 Mathematics Areas of Parallelograms and Triangles Mix Examples - Areas of Parallelograms and Triangles

Easy

Observe the given figure. Does the parallelogram $ABCD$ and the triangle $QBC$ lie on the same base and between the same parallels? If yes,write the common base and the two parallels.

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(N/A) No,the parallelogram $ABCD$ and the triangle $QBC$ do not lie on the same base and between the same parallels. For figures to lie on the same base and between the same parallels,they must share a common side as a base,and their vertices opposite to the base must lie on a line parallel to the base. In this figure,the parallelogram $ABCD$ has base $AB$ (or $CD$),while the triangle $QBC$ has base $BC$. Since they do not share a common base,the condition is not satisfied.

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Class 9

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Areas of Parallelograms and Triangles

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Mix Examples - Areas of Parallelograms and Triangles

In the figure,$ABCD$ and $AEFD$ are two parallelograms. Prove that $\operatorname{ar}(\triangle PEA) = \operatorname{ar}(\triangle QFD)$.

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In $\Delta ABC,$ $P$ and $Q$ are the points of trisection of $BC$ (i.e.,points dividing $BC$ into three equal parts). Prove that,$\operatorname{ar}(ABP) = \operatorname{ar}(APQ) = \operatorname{ar}(AQC) = \frac{1}{3} \operatorname{ar}(ABC).$

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If the medians of a $\Delta ABC$ intersect at $G$,show that $\operatorname{ar}(AGB) = \operatorname{ar}(AGC) = \operatorname{ar}(BGC) = \frac{1}{3} \operatorname{ar}(ABC)$.

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$D, E$ and $F$ are the midpoints of the sides $BC, CA$ and $AB$ respectively of $\Delta ABC$. Show that:
$(i)$ $BDEF$ is a parallelogram.
$(ii)$ $ar(DEF) = \frac{1}{4} ar(ABC)$
$(iii)$ $ar(BDEF) = \frac{1}{2} ar(ABC)$

Difficult

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$O$ is any point on the diagonal $PR$ of a parallelogram $PQRS$. Prove that $\operatorname{ar}(\triangle PSO) = \operatorname{ar}(\triangle PQO)$.

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Observe the given figure. Does the parallelogram $ABCD$ and the triangle $QBC$ lie on the same base and between the same parallels? If yes,write the common base and the two parallels.

Similar Questions

In the figure,$ABCD$ and $AEFD$ are two parallelograms. Prove that $\operatorname{ar}(\triangle PEA) = \operatorname{ar}(\triangle QFD)$.

In $\Delta ABC,$ $P$ and $Q$ are the points of trisection of $BC$ (i.e.,points dividing $BC$ into three equal parts). Prove that,$\operatorname{ar}(ABP) = \operatorname{ar}(APQ) = \operatorname{ar}(AQC) = \frac{1}{3} \operatorname{ar}(ABC).$

If the medians of a $\Delta ABC$ intersect at $G$,show that $\operatorname{ar}(AGB) = \operatorname{ar}(AGC) = \operatorname{ar}(BGC) = \frac{1}{3} \operatorname{ar}(ABC)$.

$D, E$ and $F$ are the midpoints of the sides $BC, CA$ and $AB$ respectively of $\Delta ABC$. Show that:$(i)$ $BDEF$ is a parallelogram.$(ii)$ $ar(DEF) = \frac{1}{4} ar(ABC)$$(iii)$ $ar(BDEF) = \frac{1}{2} ar(ABC)$

$O$ is any point on the diagonal $PR$ of a parallelogram $PQRS$. Prove that $\operatorname{ar}(\triangle PSO) = \operatorname{ar}(\triangle PQO)$.

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$D, E$ and $F$ are the midpoints of the sides $BC, CA$ and $AB$ respectively of $\Delta ABC$. Show that:
$(i)$ $BDEF$ is a parallelogram.
$(ii)$ $ar(DEF) = \frac{1}{4} ar(ABC)$
$(iii)$ $ar(BDEF) = \frac{1}{2} ar(ABC)$