A ladder 13 m long is placed on the ground in such a way that it touches the top of a vertical wall 12 m high.
Find the distance of the foot of the ladder from the bottom of the wall. Here, the ladder, the wall and the ground from a right-angled triangle. According to Pythagorean Theorem, xx = 5 Therefore, distance of the foot of the ladder from the bottom of the wall = 5 meters. The height of two building is 34 m and 29 m respectively.
This helps you determine the correct values to use in the different parts of the formula. The side opposite the right angle is the side labelled \(x\). When applying the Pythagorean theorem, this squared is equal to the sum of the other two sides squared.
Mathematically, this means: \(6^2 8^2 = x^2\) Which is the same as: \(100 = x^2\) Therefore, we can write: \(\beginx &= \sqrt\\ &= \bbox[border: 1px solid black; padding: 2px]\end\) Maybe you remember that in an equation like this, \(x\) could also be –10, since –10 squared is also 100.
The order of the legs isn’t important, but remember that the hypotenuse is opposite the right angle.
Now you can apply the Pythagorean theorem to write: \(x^2 y^2 = (2x)^2\) Squaring the right-hand side: \(x^2 y^2 = 4x^2\) When the problem says “the value of \(y\)”, it means you must solve for \(y\).You can still use the Pythagorean theorem in these types of problems, but you will need to be careful about the order you use the values in the formula. The side opposite the right angle has a length of 12.Therefore, we will write: \(8^2 y^2 = 12^2\) This is the same as: \(64 y^2 = 144\) Subtracting 64 from both sides: \(y^2 = 80\) Therefore: \(\beginy &= \sqrt \ &= \sqrt \ &= \bbox[border: 1px solid black; padding: 2px]\end\) In this last example, we left the answer in exact form instead of finding a decimal approximation.The easiest way to see that you should be applying this theorem is by drawing a picture of whatever situation is described.Two hikers leave a cabin at the same time, one heading due south and the other headed due west.But, the length of any side of a triangle can never be negative and therefore we only consider the positive square root.In other situations, you will be trying to find the length of one of the legs of a right triangle.Due south and due west form a right angle, and the shortest distance between any two points is a straight line.Therefore, we can apply the Pythagorean theorem and write: \(3.1^2 2.8^2 = x^2\) Here, you will need to use a calculator to simplify the left-hand side: \(17.45 = x^2\) Now use your calculator to take the square root. \(\beginx &= \sqrt \ &\approx 4.18 \text\end\) As you can see, it will be up to you to determine that a right angle is part of the situation given in the word problem.Write an expression that shows the value of \(y\) in terms of \(x\).Since no figure was given, your first step should be to draw one.