A triangular colourful scenery is made in a wall with sides 50 cm, 50 cm and 80 cm. A golden thread is to hang from the vertex so as to just reach the side 80 cm. How much length of golden thread is required?

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According to the problem, we are given a triangular colourful scenery made in a wall with sides 50 cm, 50 cm and 80 cm. A golden thread is to hang from the vertex so as to just reach the side 80 cm. We need to find the required length of golden thread.Let us draw the given information to get a better view.Let CD be the required length of the golden thread.We know that the shortest distance from any point to a line/line segment is the perpendicular distance from that point to the line/line segment.So, CD must be parallel to the side AB. We can see that the given triangle is isosceles with equal sides AC and BC. We know that the altitude, median and the perpendicular bisectors from the vertex which is formed by the intersection of two equal sides in an isosceles coincide.We know that the median passes through the midpoint of the other side. So, point D is the midpoint of the side AB.So, we get a length of BD = 12×80=40cm12×80=40cm.From the figure, we can see that the triangle BDC is a right-angled triangle with right angle at vertex D.We know that the sum of the squares of the other two sides of the right-angled triangle is equal to the square of the hypotenuse (from Pythagoras theorem).So, we get BC2=BD2+DC2BC2=BD2+DC2.⇒502=402+DC2⇒502=402+DC2.⇒2500=1600+DC2⇒2500=1600+DC2.⇒DC2=900⇒DC2=900.⇒DC=30⇒DC=30.So, we have found the required length of the golden thread as 30 cm