Advertisements
Advertisements
Question
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC2 = AD2 + BC x DE + `(1)/(4)"BC"^2`
Advertisements
Solution
We have ∠AED = 90°
∴ ∠ADE < 90° and ∠ADC > 90°
i.e. ∠ADE is acute and ∠ADC is obtuse.
In ΔADC, ∠ADC is an obtuse angle.
∴ AC2 = AD2 + DC2 + 2 x DC x DE
⇒ AC2 = AD2 + `(1/2"BC")^2 + 2 xx (1)/(2)"BC" xx "DE"`
⇒ AC2 = AD2 + `(1)/(4)"BC"^2 + "BC" xx "DE"`
⇒ AC2 = AD2 + BC x DE + `(1)/(4)"BC"^2` . ....(i)
APPEARS IN
RELATED QUESTIONS
ABCD is a rectangle whose three vertices are B (4, 0), C(4, 3) and D(0,3). The length of one of its diagonals is
(A) 5
(B) 4
(C) 3
(D) 25
Prove that the diagonals of a rectangle ABCD, with vertices A(2, -1), B(5, -1), C(5, 6) and D(2, 6), are equal and bisect each other.
In figure, ∠B of ∆ABC is an acute angle and AD ⊥ BC, prove that AC2 = AB2 + BC2 – 2BC × BD
Identify, with reason, if the following is a Pythagorean triplet.
(4, 9, 12)
Identify, with reason, if the following is a Pythagorean triplet.
(10, 24, 27)
AD is drawn perpendicular to base BC of an equilateral triangle ABC. Given BC = 10 cm, find the length of AD, correct to 1 place of decimal.
In the figure, given below, AD ⊥ BC.
Prove that: c2 = a2 + b2 - 2ax.
Diagonals of rhombus ABCD intersect each other at point O.
Prove that: OA2 + OC2 = 2AD2 - `"BD"^2/2`
If P and Q are the points on side CA and CB respectively of ΔABC, right angled at C, prove that (AQ2 + BP2) = (AB2 + PQ2)
In the given figure, BL and CM are medians of a ∆ABC right-angled at A. Prove that 4 (BL2 + CM2) = 5 BC2.
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
The length of the diagonals of rhombus are 24cm and 10cm. Find each side of the rhombus.
In an equilateral triangle ABC, the side BC is trisected at D. Prove that 9 AD2 = 7 AB2.
From a point O in the interior of aΔABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove that: AF2 + BD2 + CE2 = OA2 + OB2 + OC2 - OD2 - OE2 - OF2
In a triangle ABC right angled at C, P and Q are points of sides CA and CB respectively, which divide these sides the ratio 2 : 1.
Prove that : 9(AQ2 + BP2) = 13AB2
Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels at a speed of `(20 "km")/"hr"` and the second train travels at `(30 "km")/"hr"`. After 2 hours, what is the distance between them?
In a quadrilateral ABCD, ∠A + ∠D = 90°. Prove that AC2 + BD2 = AD2 + BC2
[Hint: Produce AB and DC to meet at E.]
The foot of a ladder is 6 m away from its wall and its top reaches a window 8 m above the ground. Find the length of the ladder.
