B Nirmala Shastry Solutions for Exercise 1: EXERCISE 10

Medians $\mathrm{CM}$ and $\mathrm{BN}$ of $△\mathrm{ABC}$ are produced to $\mathrm{P}$ and $\mathrm{Q}$ respectively such that $\mathrm{CM}=\mathrm{MP}$ and $\mathrm{BN}=\mathrm{NQ}$. Proved that the points $\mathrm{P}, \mathrm{A}, \mathrm{Q}$ are collinear and $\mathrm{PA}=\mathrm{AQ}$.

Prove that the three line segments which join the mid-points of the sides of triangle, divide it into four triangles which are congruent to each other.

In $△\mathrm{ABC}, \mathrm{M}, \mathrm{N}$ and $\mathrm{P}$ are mid-points of sides $\mathrm{AB}, \mathrm{AC} \text{and} \mathrm{BC}$ respectively. $\mathrm{X},\hspace{0.17em}\mathrm{Y},\hspace{0.17em}\text{and} \mathrm{Z}$ are mid-points of sides of $△\mathrm{MNP}$. If $\mathrm{XY}=2.5 \mathrm{cm}, \mathrm{YZ}=3.5 \mathrm{cm} \text{and} \mathrm{XZ}=4 \mathrm{cm}$ and , find the sides of $△\mathrm{ABC}$.

$\mathrm{ABC}$ is a right-angled triangle with hypotenuse $\mathrm{AC}=13 \mathrm{cm}$ and side $\mathrm{AB}=5 \mathrm{cm}$. Perpendiculars are drawn from mid-point $\mathrm{M}$ of $\mathrm{AC}$ to $\mathrm{AB}$ and $BC$. What is the perimeter of the resulting quadrilateral?

In the quadrilateral $\mathrm{ABCD}, \mathrm{AD}\parallel \mathrm{BC}. \mathrm{P}$ and $\mathrm{Q}$ are mid-points of $\mathrm{AB}$ and $\mathrm{AC}$. Prove that $\mathrm{R}$ is the mid-point of $\mathrm{DC}$.

In the quadrilateral $\mathrm{ABCD}, \mathrm{AD}\parallel \mathrm{BC}. \mathrm{P}$ and $\mathrm{Q}$ are mid-points of $\mathrm{AB}$ and $\mathrm{AC}$. Prove that $\mathrm{AD}+\mathrm{BC}=2\mathrm{PR}$.

In parallelogram $\mathrm{PQRS}, \mathrm{M}$ is the mid-point of $\mathrm{PQ}$, $\mathrm{PT}$ drawn parallel to $\mathrm{MR}$ meets $\mathrm{SR}$ at $\mathrm{N}$ and $\mathrm{QR}$ produced at $\mathrm{T}$. Prove that $\mathrm{PS}=\frac{1}{2}\mathrm{QT}$.

In parallelogram $\mathrm{PQRS}, \mathrm{M}$ is the mid-point of $\mathrm{PQ}$, $\mathrm{PT}$ drawn parallel to $\mathrm{MR}$ meets $\mathrm{SR}$ at $\mathrm{N}$ and $\mathrm{QR}$ produced at $\mathrm{T}$. Prove that $\mathrm{PT}=2\mathrm{MR}$.