Question

bridge cable length
a cable - stayed bridge is supported by several cables in tension that are attached to vertical towers. how can you determine the length of the longest cable shown? you can use the pythagorean theorem to solve problems involving triangles.
▶ step 1 draw a representation of the problem.
the height of the bridge is 36 meters.
the length is 27 meters.
▶ step 2 apply the pythagorean theorem.
( a^2 + b^2 = c^2 )
( 27^2 + 36^2 = c^2 )
( 729 + 1296 = c^2 )
( sqrt{2025} = c )
( 45 = c )
the length of the longest cable is 45 meters.
let’s explore more
a. how can you use the pythagorean theorem to determine the length of a missing leg, given the lengths of one leg and the hypotenuse?
b. ancient egyptians discovered that a triangle with side lengths 3, 4, 5 is a right triangle. how could you use this fact to solve the bridge cable length problem above?
c. using the same 3, 4, 5 relationship, can you name another set of three whole numbers that make a right triangle? use the pythagorean theorem to check your answer.

Explanation:

Part a

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). If we know one leg (\(a\) or \(b\)) and the hypotenuse (\(c\)), we can solve for the missing leg. Let's say we know leg \(a\) and hypotenuse \(c\), then we can rearrange the formula to \(b = \sqrt{c^{2}-a^{2}}\) (or \(a=\sqrt{c^{2}-b^{2}}\) if we know \(b\) and \(c\)).

Step2: Example Application

Suppose we have a right triangle with hypotenuse \(c = 5\) and one leg \(a = 3\). To find the missing leg \(b\), we substitute into the formula: \(b=\sqrt{5^{2}-3^{2}}=\sqrt{25 - 9}=\sqrt{16}=4\).

Step1: Recognize Similar Triangles

The triangle with sides 3, 4, 5 is a right triangle. The bridge cable problem involves a right triangle with legs 27 m and 36 m. Notice that \(27 = 9\times3\) and \(36=9\times4\), so the triangle in the bridge problem is similar to the 3 - 4 - 5 triangle (scaled by a factor of 9).

Step2: Use Proportionality

Since the ratio of the legs in the 3 - 4 - 5 triangle is \(3:4\), and the legs of the bridge triangle are \(27:36 = 3:4\) (dividing both by 9), the hypotenuse of the bridge triangle should be \(9\times5=45\) (since the hypotenuse of the 3 - 4 - 5 triangle is 5).

Step1: Recall Scaling of Right Triangles

If we scale the 3 - 4 - 5 triangle by a positive integer factor \(k\), we get a new right triangle with sides \(3k\), \(4k\), \(5k\). Let's choose \(k = 2\).

Step2: Check with Pythagorean Theorem

For \(k = 2\), the sides are \(3\times2 = 6\), \(4\times2=8\), \(5\times2 = 10\). Check if \(6^{2}+8^{2}=10^{2}\). Calculate \(6^{2}=36\), \(8^{2}=64\), \(10^{2}=100\). Then \(36 + 64=100\), which is true. So 6, 8, 10 is another set.

Answer:

Use \(b=\sqrt{c^{2}-a^{2}}\) (or \(a=\sqrt{c^{2}-b^{2}}\)) where \(a,b\) are legs and \(c\) is hypotenuse.

Part b