Advertisements
Advertisements
Question
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB2 + AC2 = 2AD2 + `(1)/(2)"BC"^2`
Advertisements
Solution
We have ∠AED = 90°
∴ ∠ADE < 90° and ∠ADC > 90°
i.e. ∠ADE is acute and ∠ADC is obtuse.
Adding (i) and (ii), we have
AC2 + AB2 = AD2 + BC x DE + `(1)/(4)"BC"^2 + "AD"^2 - "BC" xx "DE" + (1)/(4)"BC"^2`
⇒ AB2 + AC2 = `2"AD"^2 + (1)/(2)"BC"^2`. ....(iii)
APPEARS IN
RELATED QUESTIONS
A man goes 10 m due east and then 24 m due north. Find the distance from the starting point
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2, prove that ABC is a right triangle.
In the given figure, point T is in the interior of rectangle PQRS, Prove that, TS2 + TQ2 = TP2 + TR2 (As shown in the figure, draw seg AB || side SR and A-T-B)
Digonals of parallelogram WXYZ intersect at point O. If OY =5, find WY.
In ΔMNP, ∠MNP = 90˚, seg NQ ⊥ seg MP, MQ = 9, QP = 4, find NQ.
Find the length of diagonal of the square whose side is 8 cm.
Find the side of the square whose diagonal is `16sqrt(2)` cm.
Triangle ABC is right-angled at vertex A. Calculate the length of BC, if AB = 18 cm and AC = 24 cm.
Find the Pythagorean triplet from among the following set of numbers.
2, 6, 7
Find the Pythagorean triplet from among the following set of numbers.
4, 7, 8
The sides of the triangle are given below. Find out which one is the right-angled triangle?
11, 60, 61
A right triangle has hypotenuse p cm and one side q cm. If p - q = 1, find the length of third side of the triangle.
In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.
SHow that ∠RTP = 90°.
A man goes 18 m due east and then 24 m due north. Find the distance of his current position from the starting point?
Find the unknown side in the following triangles
Rithika buys an LED TV which has a 25 inches screen. If its height is 7 inches, how wide is the screen? Her TV cabinet is 20 inches wide. Will the TV fit into the cabinet? Give reason
If ΔABC ~ ΔPQR, `("ar" triangle "ABC")/("ar" triangle "PQR") = 9/4` and AB = 18 cm, then the length of PQ is ______.
For going to a city B from city A, there is a route via city C such that AC ⊥ CB, AC = 2x km and CB = 2(x + 7) km. It is proposed to construct a 26 km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of the highway.
Height of a pole is 8 m. Find the length of rope tied with its top from a point on the ground at a distance of 6 m from its bottom.
