In Delta ABC, M and N are the midpoints of ov | vedclass

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

Question

(B) Given that $M$ and $N$ are the midpoints of sides $AB$ and $AC$ of $\Delta ABC$ respectively.According to the Midpoint Theorem, the line segment j

Vedclass · Accepted Answer

$22.5$

Vedclass · Answer

(B) Given that $M$ and $N$ are the midpoints of sides $AB$ and $AC$ of $\Delta ABC$ respectively.According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side.Therefore, $MN \parallel BC$ and $MN = \frac{1}{2} BC$.Since $MN \parallel BC$, $\Delta AMN$ is similar to $\Delta ABC$ by the $AA$ similarity criterion ($\angle AMN = \angle ABC$ and $\angle ANM = \angle ACB$ as corresponding angles).The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.Thus, $\frac{	ext{Area}(\Delta AMN)}{	ext{Area}(\Delta ABC)} = \left( \frac{MN}{BC} ight)^2$.Substituting $MN = \frac{1}{2} BC$, we get $\frac{	ext{Area}(\Delta AMN)}{	ext{Area}(\Delta ABC)} = \left( \frac{1}{2} ight)^2 = \frac{1}{4}$.Given $	ext{Area}(\Delta ABC) = 90$, we have $	ext{Area}(\Delta AMN) = \frac{1}{4} 	imes 90 = 22.5$.

In $\Delta ABC$, $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. If the area of $\Delta ABC$ is $90$, find the area of $\Delta AMN$.

Similar Questions

Prove that if in a triangle the square of one side is equal to the sum of the squares of the other two sides,then the angle opposite the first side is a right angle.

In $\Delta PQR$,$m \angle Q = 90^\circ$. If $PQ = 5$ and $PR = 13$,find the area of $\Delta PQR$.

Given $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. If $2 m \angle P = 3 m \angle Q$ and $m \angle C = 100^{\circ}$,find $m \angle B$. (in $^{\circ}$)

In $\Delta ABC$ and $\Delta PQR$,$\angle A \cong \angle P$ and $\angle B \cong \angle R$. If $AB = 8$,$PQ = 7.5$ and $AC = 6$,find $PR$.

In $\Delta XYZ$,$m \angle Y = 90^\circ$ and $\overline{YM}$ is an altitude to the hypotenuse $\overline{XZ}$. If $XM = 5$ and $ZM = 4$,find $YZ$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module