In Delta ABC,overline{AB} cong overline{AC} a | vedclass

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

Question

(A) In $\Delta ABC$,$\overline{AD}$ is a median to the base $\overline{BC}$.Since $\Delta ABC$ is an isosceles triangle with $\overline{AB} \cong \ove

Vedclass · Accepted Answer

$60$

Vedclass · Answer

(A) In $\Delta ABC$,$\overline{AD}$ is a median to the base $\overline{BC}$.Since $\Delta ABC$ is an isosceles triangle with $\overline{AB} \cong \overline{AC}$,the median $\overline{AD}$ to the base is also the altitude to the base.Therefore,$\overline{AD} \perp \overline{BC}$ and $\angle ADB = 90^{\circ}$.Let $AB = AC = x$. The perimeter of $\Delta ABC$ is $AB + AC + BC = 36$.$x + x + BC = 36 \implies BC = 36 - 2x$.Since $D$ is the midpoint of $\overline{BC}$,$BD = \frac{BC}{2} = \frac{36 - 2x}{2} = 18 - x$.In right-angled $\Delta ABD$,by the Pythagorean theorem:$AB^2 = AD^2 + BD^2$$x^2 = 12^2 + (18 - x)^2$$x^2 = 144 + 324 - 36x + x^2$$36x = 468$$x = 13$.Thus,$BD = 18 - 13 = 5$,so $BC = 2 	imes 5 = 10$.The area of $\Delta ABC = \frac{1}{2} 	imes 	ext{base} 	imes 	ext{height} = \frac{1}{2} 	imes BC 	imes AD$.Area $= \frac{1}{2} 	imes 10 	imes 12 = 60$.

In $\Delta ABC$,$\overline{AB} \cong \overline{AC}$ and $\overline{AD}$ is a median. If $AD = 12$ and the perimeter of $\Delta ABC$ is $36$,find the area of $\Delta ABC$.

Similar Questions

In $\Delta ABC$,$m \angle A = m \angle B + m \angle C$. If $AB = 7$ and $BC = 25$,the perimeter of $\Delta ABC = ........$

The perimeter of rhombus $ABCD$ is $68$. If $AC = 30$,find $BD$.

In the figure,$l \parallel m$ and line segments $AB, CD$ and $EF$ are concurrent at point $P$. Prove that $\frac{AE}{BF} = \frac{AC}{BD} = \frac{CE}{FD}$.

In the figure,if $\triangle ABC \sim \triangle DEF$ and their side lengths (in $cm$) are as marked,find the lengths of the sides of each triangle.

Prove that the area of the semicircle drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the semicircles drawn on the other two sides of the triangle.

Vedclass Test Series

Exam Paper Generator

Online Exam Module