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The sides of a triangle are 3x+4y, 4x+3y and 5x+5y where x, y > 0 then the triangle is
Let a = 3x + 4y, b = 4x + 3y and c = 5x + 5y as x, y > 0, c = 5x + 5y is the largest side
∴ C is the largest angle . Now
∴ C is obtuse angle Þ DABC is obtuse angled
In a triangle with sides a, b, c, r1 > r2 > r3 (which are the exradii) then
The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a , is
In a triangle ABC, medians AD and BE are drawn. If AD = 4, then the area of the Δ ABC is
[∴ Median of a D divides it into two Δ's of equal area.]
If in a ΔABC then the sides a, b and c
a[cos C +1] + c[cos A +1] = 3b
(a + c) + (a cos C + c cos B) = 3b
a + c + b = 3b or a + c = 2b or a,b,c arein A.P.
The sides of a triangle are sin α, cos α and Then the greatest angle of the triangle is
Let a = sin a,b = cosa and
Clearly a and b < 1 but c > 1 as sina > 0 and cosa > 0
∴ c is the greatest side and greatest angle is C
∴ C = 120°
A person stan ding on t he bank of a river obser ves that the angle of elevation of the top of a tree on the opposite bank of the river is 60° and when he retires 40 meters away from the tree the angle of elevation becomes 30°. The breadth of the river is
From the figure
In a triangle ABC, let If r is the inradius and R is the circumradius of the triangle ABC, then 2 (r + R) equals
⇒ c= 2R (∵ ∠C = 90°)
(∵ ∠C = 90°)
⇒ 2r +c = a+b ⇒ 2r +2 R = a+b (using c = 2R)
If in a ΔABC , the altitudes from the vertices A, B, C on opposite sides are in H.P, then sin A, sin B, sin C are in
⇒ a, b, c are in A.P.
⇒ KsinA, K sin B, K sin C are in AP
⇒ sinA, sinB, sinC are in A.P.
A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that AB (= a) subtends an angle of 60° at the foot of the tower, and the angle of elevation of the top of the tower from A or B is 30°.
The height of the tower is
In the Δ AOB, ∠ AOB = 60°, and ∠ OBA = ∠ OAB (since OA = OB = AB radius of same circle). ∴ D AOB is a equilateral triangle. Let the height of tower is h m. Given distance between two points A & B lie on boundary of circular park, subtends an angle of 60° at the foot of the tower is AB i.e. AB = a. A tower OC stands at the centre of a circular park. Angle of elevation of the top of the tower from A and B is 30°
AB is a vertical pole with B at the ground level and A at the top. A man finds that the angle of elevation of the point A from a certain point C on the ground is 60°. He moves away from the pole along the line BC to a point D such that CD = 7 m. From D the angle of elevation of the point A is 45°. Then the height of the pole is
For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is
If O is centre of polygon and AB is one of the side, then by figure
n = 3, 4, 6 respectively.
A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is 45° . It flies off horizontally straight away from the point O. After one second, the elevation of the bird from O is reduced to 30° . Then the speed (in m/s) of the bird is
Let the speed be y m/sec.
Let AC be the vertical pole of height 20 m.
Let O be the point on the ground such that ∠AOC = 45° Let OC = x
If the angles of elevation of the top of a tower from three collinear points A, B and C, on a line leading to the foot of the tower, are 30°, 45° and 60° respectively, then the ratio, AB : BC, is :
∵ PB bisects ÐAPC, therefore AB : BC = PA : PC