square ABCD is a trapezium in which overline{ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given that $\square ABCD$ is a trapezium with $\overline{AB} \parallel \overline{CD}$.We are given $AM 	imes NC = BN 	imes MD$,which can be rewr

Vedclass · Accepted Answer

$\frac{2}{5}$

Vedclass · Answer

(A) Given that $\square ABCD$ is a trapezium with $\overline{AB} \parallel \overline{CD}$.We are given $AM 	imes NC = BN 	imes MD$,which can be rewritten as $\frac{AM}{MD} = \frac{BN}{NC}$.Since $\frac{AM}{MD} = \frac{2}{3}$,it follows that $\frac{BN}{NC} = \frac{2}{3}$.By the Basic Proportionality Theorem (Thales Theorem) or its converse,since $\frac{AM}{MD} = \frac{BN}{NC}$,the line segment $\overline{MN}$ must be parallel to the bases $\overline{AB}$ and $\overline{CD}$.Let $\overline{MN} \parallel \overline{AB} \parallel \overline{CD}$.In $	riangle ADC$,since $\overline{MO} \parallel \overline{CD}$,by the intercept theorem (or similarity of triangles $	riangle AMO \sim 	riangle ADC$),we have $\frac{AO}{AC} = \frac{AM}{AD}$.Given $\frac{AM}{MD} = \frac{2}{3}$,let $AM = 2k$ and $MD = 3k$. Then $AD = AM + MD = 2k + 3k = 5k$.Therefore,$\frac{AO}{AC} = \frac{AM}{AD} = \frac{2k}{5k} = \frac{2}{5}$.

Similar Questions

In $\Delta ABC$,$m \angle B = 90^{\circ}$. If $AB = 6$ and $AC = 10$,find the area of $\Delta ABC$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$,$\overline{BM}$ is an altitude to the hypotenuse $AC$,and $AM < CM$. If $BM = 6$ and $AC = 13$,find $AB$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM : MB = 4 : 13$ and $AC = 20.4$,find $AN$.

In $\Delta ABC$,$\overline{AD}$ and $\overline{BE}$ are medians. Line $m$ through $D$ and parallel to $\overline{BE}$ intersects $\overline{AC}$ at $K$. If $EK = 2.5$,then $AC = \ldots$

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BE}$ is a median. If $AB = 3.6$ and $BC = 4.8$,find $BE$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module