Advertisements
Advertisements
Question
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB2 + AC2 = 2(AD2 + CD2)
Advertisements
Solution
We have ∠AED = 90°
∴ ∠ADE < 90° and ∠ADC > 90°
i.e. ∠ADE is acute and ∠ADC is obtuse.
From (iii), we have
AB2 + AC2 = `2"AD"^2 + (1)/(2)"BC"^2`
⇒ AB2 + AC2 = `2"AD"^2 + (1)/(2)(2 xx "CD")^2`
⇒ AB2 + AC2 = `2"AD"^2 + (1)/(2) xx 4"CD"^2`
⇒ AB2 + AC2 = 2AD2 + 2CD2
⇒ AB2 + AC2 = 2(AD2 + CD2).
APPEARS IN
RELATED QUESTIONS
The perpendicular AD on the base BC of a ∆ABC intersects BC at D so that DB = 3 CD. Prove that `2"AB"^2 = 2"AC"^2 + "BC"^2`
ABC is a right triangle right-angled at C. Let BC = a, CA = b, AB = c and let p be the length of perpendicular from C on AB, prove that
(i) cp = ab
`(ii) 1/p^2=1/a^2+1/b^2`
In an equilateral triangle ABC, D is a point on side BC such that BD = `1/3BC` . Prove that 9 AD2 = 7 AB2
Which of the following can be the sides of a right triangle?
2.5 cm, 6.5 cm, 6 cm
In the case of right-angled triangles, identify the right angles.
If the sides of the triangle are in the ratio 1: `sqrt2`: 1, show that is a right-angled triangle.
O is any point inside a rectangle ABCD.
Prove that: OB2 + OD2 = OC2 + OA2.
If the angles of a triangle are 30°, 60°, and 90°, then shown that the side opposite to 30° is half of the hypotenuse, and the side opposite to 60° is `sqrt(3)/2` times of the hypotenuse.
In the given figure, angle ACP = ∠BDP = 90°, AC = 12 m, BD = 9 m and PA= PB = 15 m. Find:
(i) CP
(ii) PD
(iii) CD
In the figure below, find the value of 'x'.
Two poles of height 9m and 14m stand on a plane ground. If the distance between their 12m, find the distance between their tops.
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC2 - AB2 = 2BC x ED
PQR is an isosceles triangle with PQ = PR = 10 cm and QR = 12 cm. Find the length of the perpendicular from P to QR.
∆ABC is right-angled at C. If AC = 5 cm and BC = 12 cm. find the length of AB.
A man goes 18 m due east and then 24 m due north. Find the distance of his current position from the starting point?
Choose the correct alternative:
If length of sides of a triangle are a, b, c and a2 + b2 = c2, then which type of triangle it is?
In a right angled triangle, if length of hypotenuse is 25 cm and height is 7 cm, then what is the length of its base?
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.
The hypotenuse (in cm) of a right angled triangle is 6 cm more than twice the length of the shortest side. If the length of third side is 6 cm less than thrice the length of shortest side, then find the dimensions of the triangle.
Points A and B are on the opposite edges of a pond as shown in the following figure. To find the distance between the two points, the surveyor makes a right-angled triangle as shown. Find the distance AB.
Two poles of 10 m and 15 m stand upright on a plane ground. If the distance between the tops is 13 m, find the distance between their feet.
