Given:$ABC$ is a triangle in which altitudes $BE$ and $CF$ to sides $AC$ and $AB$ are equal.To do:We have to show that(i) $\triangle ABE \cong \triangle ACF$(ii) $AB=AC$, i.e., $ABC$ is an isosceles triangle.Solution:(i) We know that, If two angles of a triangle with a non-included side are equal to ... Read More

Given:$ABC$ and $DBC$ are two isosceles triangles on the same base $BC$.To do: We have to show that $\angle ABD=\angle ACD$.Solution:Let us consider $\triangle ABD$ and $\triangle ACD$We know that, Side-Side-Side congruence rule states that if three sides of one triangle are equal to three corresponding sides of another triangle, then ... Read More

Given:$ABC$ is an isosceles triangle in which altitudes $BE$ and $CF$ are drawn to equal sides $AC$ and $AB$ respectively.To do:We have to show that these altitudes are equal.Solution:Let us consider $\triangle AEB$ and $\triangle AFC$We know that, From Angle-Angle-Side:if two angles of a triangle with a non-included side are ... Read More

Given:In a parallelogram \( \mathrm{ABCD}, \mathrm{E} \) and \( \mathrm{F} \) are the mid-points of sides \( \mathrm{AB} \) and \( \mathrm{CD} \) respectively.To do:We have to show that the line segments $AF$ and \( \mathrm{EC} \) trisect the diagonal $BD$. Solution:$A B C D$ is a parallelogram.We know that, Opposite ... Read More

Given:\( \mathrm{ABCD} \) is a trapezium in which \( \mathrm{AB} \| \mathrm{DC}, \mathrm{BD} \) is a diagonal and \( \mathrm{E} \) is the mid-point of \( \mathrm{AD} \). A line is drawn through E parallel to \( \mathrm{AB} \) intersecting \( \mathrm{BC} \) at \( \mathrm{F} \)To do:We have to ... Read More

Given:\( \mathrm{ABC} \) is a triangle right angled at \( \mathrm{C} \). A line through the mid-point \( \mathrm{M} \) of hypotenuse \( \mathrm{AB} \) and parallel to \( \mathrm{BC} \) intersects \( \mathrm{AC} \) at \( \mathrm{D} \).To do:We have to show that (i) \( \mathrm{D} \) is the mid-point ... Read More

Given:$ABCD$ is a quadrilateral in which \( P, Q, R \) and \( S \) are mid-points of the sides \( \mathrm{AB}, \mathrm{BC}, \mathrm{CD} \) and \( \mathrm{DA} \). $AC$ is a diagonal.To do:We have to show that(i) \( \mathrm{SR} \| \mathrm{AC} \) and \( \mathrm{SR}=\frac{1}{2} \mathrm{AC} \)(ii) \( \mathrm{PQ}=\mathrm{SR} ... Read More

Given:\( \mathrm{ABCD} \) is a rectangle and \( \mathrm{P}, \mathrm{Q}, \mathrm{R} \) and \( \mathrm{S} \) are mid-points of the sides \( \mathrm{AB}, \mathrm{BC}, \mathrm{CD} \) and \( \mathrm{DA} \) respectively.To do:We have to show that the quadrilateral \( \mathrm{PQRS} \) is a rhombus. Solution:$\angle A=\angle \mathrm{B}=\angle \mathrm{C}=\angle \mathrm{D}=90^{\circ}$$\mathrm{AD}=\mathrm{BC}$ and $\mathrm{AB}=\mathrm{CD}$$P, ... Read More

Given:\( \mathrm{ABCD} \) is a rhombus and \( \mathrm{P}, \mathrm{Q}, \mathrm{R} \) and \( \mathrm{S} \) are the mid-points of the sides \( \mathrm{AB}, \mathrm{BC}, \mathrm{CD} \) and DA respectively. To do:We have to show that the quadrilateral \( \mathrm{PQRS} \) is a rectangle. Solution:Join $PQ, QR, RS, PS, AC$ and $BD$.In ... Read More

Given:\( \mathrm{ABCD} \) is a trapezium in which \( \mathrm{AB} \| \mathrm{CD} \) and \( \mathrm{AD}=\mathrm{BC} \).To do :We have to show that(i) \( \angle \mathrm{A}=\angle \mathrm{B} \)(ii) \( \angle \mathrm{C}=\angle \mathrm{D} \)(iii) \( \triangle \mathrm{ABC} \equiv \triangle \mathrm{BAD} \)(iv) diagonal \( \mathrm{AC}= \) diagonal \( \mathrm{BD} \)Solution :  Extend ... Read More