Advertisements
Advertisements
Question
If the sides of the triangle are in the ratio 1: `sqrt2`: 1, show that is a right-angled triangle.
Advertisements
Solution
Let, the sides of the triangle be, x: `sqrt2`x and x.
AB2 + BC2 = x2 +x2 = 2x2
AC2 = `(sqrt2 x)^2` = 2x2
AB2 + BC2 = AC2
Conversely, if in any triangle, the square on the largest side of the triangle is equal to the sum of the squares on remaining two sides, then the triangle is a right-angled triangle and the angle opposite to the largest side is a right-angle.
Therefore, Δ ABC is a right-angled triangle.
APPEARS IN
RELATED QUESTIONS
P and Q are the mid-points of the sides CA and CB respectively of a ∆ABC, right angled at C. Prove that:
`(i) 4AQ^2 = 4AC^2 + BC^2`
`(ii) 4BP^2 = 4BC^2 + AC^2`
`(iii) (4AQ^2 + BP^2 ) = 5AB^2`
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2, prove that ABC is a right triangle.
A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.
Tick the correct answer and justify: In ΔABC, AB = `6sqrt3` cm, AC = 12 cm and BC = 6 cm.
The angle B is:
Which of the following can be the sides of a right triangle?
2 cm, 2 cm, 5 cm
In the case of right-angled triangles, identify the right angles.
Identify, with reason, if the following is a Pythagorean triplet.
(3, 5, 4)
In ∆ABC, AB = 10, AC = 7, BC = 9, then find the length of the median drawn from point C to side AB.
In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1: 3.
Prove that : 2AC2 = 2AB2 + BC2
The sides of a certain triangle is given below. Find, which of them is right-triangle
6 m, 9 m, and 13 m
In the figure below, find the value of 'x'.
Find the length of the perpendicular of a triangle whose base is 5cm and the hypotenuse is 13cm. Also, find its area.
The length of the diagonals of rhombus are 24cm and 10cm. Find each side of the rhombus.
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB2 = AD2 - BC x CE + `(1)/(4)"BC"^2`
Find the unknown side in the following triangles
For going to a city B from city A, there is a route via city C such that AC ⊥ CB, AC = 2x km and CB = 2(x + 7) km. It is proposed to construct a 26 km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of the highway.
Lengths of sides of a triangle are 3 cm, 4 cm and 5 cm. The triangle is ______.
Two rectangles are congruent, if they have same ______ and ______.
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.
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.
