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✅Question :

The length of the base of this triangle is given by b-√(l^2-36). A second larger triangle is constructed with the same height, its base is twice as long. Find an expression for the length of the hypotenuse of the larger triangle (L) in terms of l. Show all steps of working it out.

✅Answer :

Let’s start by considering the properties of the two triangles and their bases.

**First Triangle**:- Base: b – √(l^2 – 36)
- Height: h (not specified, but assumed to be constant for both triangles)

**Second Triangle**:- Base: 2b (twice as long as the base of the first triangle)
- Height: h (same height as the first triangle)

Both triangles have the same height, so their hypotenuses are related by the Pythagorean theorem.

For the first triangle: Hypotenuse of the first triangle (h1) = √[(base)^2 + (height)^2]

For the second triangle: Hypotenuse of the second triangle (h2) = √[(2b)^2 + (height)^2]

Since h1 = h2 (both triangles share the same height), we can set up an equation:

√[(b – √(l^2 – 36))^2 + h^2] = √[(2b)^2 + h^2]

Now, let’s solve for b in terms of h and l:

- Square both sides of the equation to eliminate the square roots.
- Expand and simplify the expressions.
- Solve for b.

Step 1: [(b – √(l^2 – 36))^2 + h^2] = (2b)^2 + h^2

Step 2: b^2 – 2b√(l^2 – 36) + (l^2 – 36) + h^2 = 4b^2 + h^2

Step 3: Collect the terms involving b on one side of the equation: b^2 – 4b^2 = 2b√(l^2 – 36) + l^2 – 36

Simplify the left side: -3b^2 = 2b√(l^2 – 36) + l^2 – 36

Divide both sides by -3b (assuming b ≠ 0): b = [(l^2 – 36) + 2b√(l^2 – 36)] / (3b)

Now, we can express b in terms of l: 1 = [(l^2 – 36) / (3b)] + 2√(l^2 – 36) / (3b)

Solve for 1 – [(l^2 – 36) / (3b)] to get an expression for b: 1 – [(l^2 – 36) / (3b)] = 2√(l^2 – 36) / (3b)

Now, we can find the expression for the length of the hypotenuse of the larger triangle (L) in terms of l: L = √[(2b)^2 + h^2] L = √[(2[(l^2 – 36) + 2b√(l^2 – 36)] / (3b))^2 + h^2]

This expression for L in terms of l is quite complex and involves b, which itself is defined in terms of l. It may not be possible to further simplify this expression without additional information or numerical values for b, h, and l.

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