Advertisements
Advertisements
प्रश्न
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 = AE2 + CD2 + BF2
Advertisements
उत्तर
In right triangles ODB and ODC, we have
OB2 = OD2 + BD2
OC2 = OD2 + CD2
∴ OB2 - OC2 = (OD2 + BD2) - (OD2 + CD2)
⇒ OB2 - OC2 = BD2 - CD2 ....(i)
Similarly, we have
OC2 - OA2 = CE2 - AE2 ....(ii)
OA2 - OB2 = AF2 - BF2 ....(iii)
Adding (i), (ii) and (iii), we get
(OB2 - OC2) + (OC2 - OA2) + (OA2 - OB2) = (BD2 - CD2) + (CE2 - AE2) + (AF2 - BF2)
⇒ (BD2 + CE2 + AF2) - (AE2 + CD2 + BF2) = 0
⇒ AF2 + BD2 + CE2 = AE2 + CD2 + BF2.
APPEARS IN
संबंधित प्रश्न
The points A(4, 7), B(p, 3) and C(7, 3) are the vertices of a right traingle ,right-angled at B. Find the values of p.
In Figure, ABD is a triangle right angled at A and AC ⊥ BD. Show that AD2 = BD × CD
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2, prove that ABC is a right triangle.
In the following figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that
(i) OA2 + OB2 + OC2 − OD2 − OE2 − OF2 = AF2 + BD2 + CE2
(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2
In the given figure, ABC is a triangle in which ∠ABC < 90° and AD ⊥ BC. Prove that AC2 = AB2 + BC2 − 2BC.BD.
In the given figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:
`"AC"^2 = "AD"^2 + "BC"."DM" + (("BC")/2)^2`
Identify, with reason, if the following is a Pythagorean triplet.
(4, 9, 12)
Walls of two buildings on either side of a street are parallel to each other. A ladder 5.8 m long is placed on the street such that its top just reaches the window of a building at the height of 4 m. On turning the ladder over to the other side of the street, its top touches the window of the other building at a height 4.2 m. Find the width of the street.
Pranali and Prasad started walking to the East and to the North respectively, from the same point and at the same speed. After 2 hours distance between them was \[15\sqrt{2}\]
km. Find their speed per hour.
In triangle ABC, given below, AB = 8 cm, BC = 6 cm and AC = 3 cm. Calculate the length of OC.
Show that the triangle ABC is a right-angled triangle; if: AB = 9 cm, BC = 40 cm and AC = 41 cm
The sides of the triangle are given below. Find out which one is the right-angled triangle?
1.5, 1.6, 1.7
From the given figure, find the length of hypotenuse AC and the perimeter of ∆ABC.
In ΔABC, AD is perpendicular to BC. Prove that: AB2 + CD2 = AC2 + BD2
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC2 - AB2 = 2BC x ED
In the given figure. PQ = PS, P =R = 90°. RS = 20 cm and QR = 21 cm. Find the length of PQ correct to two decimal places.
To get from point A to point B you must avoid walking through a pond. You must walk 34 m south and 41 m east. To the nearest meter, how many meters would be saved if it were possible to make a way through the pond?
If ΔABC ~ ΔPQR, `("ar" triangle "ABC")/("ar" triangle "PQR") = 9/4` and AB = 18 cm, then the length of PQ is ______.
Sides AB and BE of a right triangle, right-angled at B are of lengths 16 cm and 8 cm respectively. The length of the side of largest square FDGB that can be inscribed in the triangle ABE is ______.
The perimeter of the rectangle whose length is 60 cm and a diagonal is 61 cm is ______.
