Download App

Triangles — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTriangles3 Marks
Question
In $\triangle\text{ABC},\text{D}$ is the midpoint of BC and $\text{AE}\perp\text{BC}.$ If $\text{AC}>\text{AB},$ show that.
$\text{AB}^2=\text{AD}^2-\text{BC}.\text{DE}+\frac{1}{4}\text{BC}^2.$
✓

Answer

Given: $\triangle\text{ABC}$ in which D is the midpoint of BC.
$\text{AE}\perp\text{BC}$ and AC > AB.
Then BD = CD and $\angle\text{AED}=90^\circ,$
Then, $\angle\text{ADE}<90^\circ$ and $\angle\text{ADC}>90^\circ

In $\triangle\text{AED},$ $\angle\text{AED}=90^\circ$
$\therefore\text{AD}^2=\text{AE}^2+\text{DE}^2$
$\Rightarrow\text{AE}^2=\big(\text{AD}^2-\text{DE}^2\big)\dots(1)$
In $\triangle\text{AEB},\angle\text{AEB}=90^\circ$
$\therefore \text{AB}^2=\text{AE}^2+\text{BE}^2\dots(2)$
Putting value of $AE^2$ from (1) in (2),
we get $\therefore\text{AB}^2=\big(\text{AD}^2-\text{DE}^2\big)+\text{BE}^2$
$=\big(\text{AD}^2-\text{DE}^2\big)+\big(\text{BD}^2-\text{DE}^2\big)\Big[\text{But}\text{ BD}=\frac{1}{2}\text{BC}\Big]$
$=\text{AD}^2-\text{DE}^2+\Big(\frac{1}{2}\text{BC}-\text{DE}\Big)^2$
​​​​​​​ $=\text{AD}^2-\text{DE}^2+\frac{1}{4}\text{BC}^2+\text{DE}^2-\text{BC.DE}$ $\text{AB}^2=\text{AD}^2-\text{BC},\text{DE}+\frac{1}{4}\text{BC}^2$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Triangles→Triangles — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

An iron pole consisting of a cylindrical portion $110\ cm$ high and of base diameter $12\ cm$ is surmounted by a cone $9\ cm$ high. Find the mass of the pole, given that $1\ cm^3$​​​​​​​ of iron has 8 gram mass approximately. $\Big(\text{Use}:\ \pi=\frac{355}{115}\Big)$
→
In the following, determine whether the given quadratic equation have real root and if so, find the root:
$x^2 - 2x + 1 = 0$
→
Draw a right triangle ABC in which $A B=6 cm$, $B C=8 cm$ and $\angle B=90^{\circ}$. Draw $B D$ perpendicular from $B$ on $A C$ and draw a circle passing through the points $B, C$ and $D$. Construct tangents from $A$ to this circle.
→
Find whether 55 is a term of the AP $7, 10, 13,---$ or not. If yes, find which term it is.
→
Is 184 a term of the AP 3, 7, 11, 15, ....?
→
Find the roots of the following quadratic equation (if they exist) by the method of completing the square.
$\sqrt{2}\text{x}^2-3\text{x}-2\sqrt{2}=0$
→
Find the values of $y$ for which the distance between the points $P(2,-3)$ and $Q(10, y)$ is 10 units.
→
If PT is a tangent to a circle with centre O and PQ is a chord of the circle such that $\angle\text{QPT} = 70^\circ$ then find the measure of $\angle\text{POQ}.$
→
Find the mean using the step deviation method.
Class0-1010-2020-3030-4040-50
Frequency61015910
→
To fill a swimming pool two pipes are used. If the pipe of larger diameter used for $4$ hours and the pipe of smaller diameter for $9$ hours, only half of the pool can be filled. Find how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes $10$ hours more than the pipe of larger diameter to fill the pool.
→