In above triangular prism,ab=3cm,ac=4cm,bc=5cm and cd12 cm,show that triangle ABC is a right-angled triangle

Well, triangle ABC seems to have some interesting side lengths! Let's see if we can uncover its true nature.

Now, since bc = 5cm, we have b^2 + c^2 = 5^2 according to the Pythagorean theorem. However, we still need to figure out the values of b and c to proceed.

To find out b, we can use the formula for the length of a line segment (in this case, ab): ab^2 = ac^2 + bc^2. Substituting the known values, we get 3^2 = 4^2 + b^2. Solving this equation, we find b = √5.

Now let's find c. Using the formula ac^2 = bc^2 + ab^2, we can plug in the values: 4^2 = 5^2 + 3^2. Solving for c, we get c = √16 = 4.

So, b = √5 and c = 4, which makes triangle ABC a √5-4-5 right-angled triangle! Now that we have unraveled this geometric mystery, I hope you're not too "tri"ed from the process!

To determine if triangle ABC is a right-angled triangle, we need to check if the square of the length of the longest side is equal to the sum of the squares of the other two sides.

We are given that:
ab = 3 cm
ac = 4 cm
bc = 5 cm
cd = 12 cm

First, let's find the length of side AB. We can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The longest side in triangle ABC is bc, so it would be the hypotenuse if triangle ABC is a right-angled triangle.

Using the Pythagorean theorem:
bc^2 = ab^2 + ac^2

Substituting the given values:
(5 cm)^2 = (3 cm)^2 + (4 cm)^2
25 cm^2 = 9 cm^2 + 16 cm^2
25 cm^2 = 25 cm^2

Since both sides of the equation are equal, we conclude that triangle ABC is a right-angled triangle.

To determine if triangle ABC is a right-angled triangle, we need to check if the square of the length of one side is equal to the sum of the squares of the other two sides. In this case, we can use the Pythagorean theorem to determine if triangle ABC is right-angled.

The Pythagorean theorem states that in a right-angled triangle, with sides of lengths a, b, and c, where c is the hypotenuse (the side opposite the right angle), the relationship between the sides is given by:

a^2 + b^2 = c^2

In our case, we have a triangle ABC with sides of lengths ab, ac, and bc. So we need to check if:

ab^2 + ac^2 = bc^2

Plugging in the given values, we have:

(3cm)^2 + (4cm)^2 = (5cm)^2

Simplifying this equation, we get:

9cm^2 + 16cm^2 = 25cm^2

25cm^2 = 25cm^2

Since both sides of the equation are equal, we can conclude that triangle ABC is a right-angled triangle.