Advertisements
Advertisements
प्रश्न
In a ΔABC, ∠CAB is an obtuse angle. P is the circumcentre of ∆ABC. Prove that ∠CAB – ∠PBC = 90°.
Advertisements
उत्तर
Given: ∠CAB is an obtuse angle and P is the circumcentre of ΔABC.
Construction: Draw BD as diameter, join AD.
Proof: ∠CAD = ∠CBD ......[Angles on same arc]
⇒ ∠CAD = ∠CBP ......(i)
Also, ∠BAD = 90° ......(ii) [Angle in semi-circle]
Now, from figure,
∠CAB = ∠CAD + ∠DAB
⇒ ∠CAB = ∠CBP + 90° ......[Using (i) and (ii)]
⇒ ∠CAB – ∠CBP = 90°
or ∠CAB – ∠PBC = 90°.
Hence proved.
APPEARS IN
संबंधित प्रश्न
If the sides of a triangle are 3 cm, 4 cm, and 6 cm long, determine whether the triangle is a right-angled triangle.
The sides of triangle is given below. Determine it is right triangle or not.
a = 8 cm, b = 10 cm and c = 6 cm
A man goes 15 metres due west and then 8 metres due north. How far is he from the starting point?
A ladder 17 m long reaches a window of a building 15 m above the ground. Find the distance of the foot of the ladder from the building.
In an isosceles triangle ABC, AB = AC = 25 cm, BC = 14 cm. Calculate the altitude from A on BC.
The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its tip reach?
ABCD is a square. F is the mid-point of AB. BE is one third of BC. If the area of ΔFBE = 108 cm2, find the length of AC.
In a ΔABC, AB = BC = CA = 2a and AD ⊥ BC. Prove that
(i) AD = a`sqrt3`
(ii) Area (ΔABC) = `sqrt3` a2
The lengths of the diagonals of a rhombus are 24 cm and 10 cm. Find each side of the rhombus.
In right-angled triangle ABC in which ∠C = 90°, if D is the mid-point of BC, prove that AB2 = 4AD2 – 3AC2.
In a quadrilateral ABCD, ∠B = 90°, AD2 = AB2 + BC2 + CD2, prove that ∠ACD = 90°.
In an equilateral triangle with side a, prove that area = `sqrt(3)/4 a^2`.
A man goes 12 m due south and then 35 m due west. How far is he from the starting point?
From given figure, In ∆ABC, AB ⊥ BC, AB = BC then m∠A = ?
From given figure, In ∆ABC, AB ⊥ BC, AB = BC, AC = `2sqrt(2)` then `l` (AB) = ?
From given figure, In ∆ABC, AB ⊥ BC, AB = BC, AC = `5sqrt(2)`, then what is the height of ∆ABC?
Find the height of an equilateral triangle having side 4 cm?
Find the altitude of an equilateral triangle of side 8 cm.
