top button
Flag Notify
    Connect to us
      Facebook Login
      Site Registration Why to Join

    Get Free Puzzle Updates

Facebook Login
Site Registration

Find the length of hypotenuse?

+1 vote
74 views

Triangle ABC is right-angled at B. D is a point on AB such that Angle BCD = Angle DCA. E is a point on BC such that Angle BAE = Angle EAC.

If AE = 9 cm and CD = 8 root 2 cm then find the length of AC.

Find length of AC

posted May 21, 2014 by Pardeep Kohli

Share this puzzle
Facebook Share Button Twitter Share Button Google+ Share Button LinkedIn Share Button Multiple Social Share Button

1 Solution

0 votes

Let angleBAE = x, and angleBCD = y.

AB = 9 cos x = AC cos 2x.
BC = 8 root 2 cos y = AC cos 2y.

Eliminating AC:
(9 cos x) / (cos 2x) = (8root 2 cos y) / (cos 2y).

y = 45° − x. Also, 2y = 90° − 2x, and so cos 2y = sin 2x. Therefore:
(9 cos x) / (cos 2x) = (8root 2 cos (45°−x)) / (sin 2x).

Using trigonometric identity cos(a − b) = cos a · cos b + sin a · sin b:
cos (45°−x) = (cos x + sin x) /root 2.

Rearranging: tan 2x = 8(cos x + sin x) / 9 cos x.

Using trigonometric identity tan 2a = 2 tan a / (1 − tan2a), and letting t = tan x:
2t / (1 − t2) = 8(1 + t)/9.
Therefore 9t/4 = (1 + t)(1 − t2) = 1 + t − t2 − t3.
Hence t3 + t2 + (5/4)t − 1 = 0.

By inspection, one root is t = 1/2.
Therefore (t − 1/2)(t2 + 3t/2 + 2) = 0.
The quadratic factor has no real roots (since (3/2)2 − 4·1·2 < 0), and so t = 1/2 is the only real root.

AC = 9 · cos x / cos 2x.

Using trigonometric identities cos x = 1 / square root(1 + t2), cos 2x = (1 − t2)/(1 + t2):
AC = 9 · square root(1 + t2) / (1 − t2).

Therefore AC = 9 · (root 5/2) / (3/4) = 6root 5 cm.

solution Aug 18, 2014 by anonymous



Similar Puzzles
+1 vote

Find the Area of the Right Angled Triangle whose hypotenuse is 11 cm and perpendicular from the right angle vertex on the hypotenuse is 6 cm.

0 votes

In the figure, semicircles are constructed on the hypotenuse and legs of a right angle triangle. If S1, S2 are the area of shaded regions and S is the area of right angled triangle, which of the following is true and why ?

  1. S1 + S2 = S
  2. S1 + S2 = 2S
  3. S1 + S2 = 3S

enter image description here

0 votes

ABC has a right angle at B with AC=60, AE=52 and CD=39. Find the length of DE?

length of DE

–1 vote

If the length of two sides of a right angled triangle measured in inches are prime numbers, and it has two adjacent sides of 12 and 13 inches long, how long must the third side be?

+1 vote

Nine unit circles are packed into a square, tangent to their neighbors and to the square. What is the length of the longest smooth path connecting two opposite corners of the square?

nine squares

Assumptions:
- The path must be continuous and follow the lines in the diagram; that is, it must be made up of portions of either the circles or the outside square.
- The path may not change direction suddenly.
- The path may not contain any loops.
- The path may not touch or cross itself at any point.

Contact Us
+91 9880187415
sales@queryhome.net
support@queryhome.net
#280, 3rd floor, 5th Main
6th Sector, HSR Layout
Bangalore-560102
Karnataka INDIA.
QUERY HOME
...