Question

XYZ is a right triangle, right angled at "Y". If YN is a perpendicular to XZ, then prove that 1/(1N ^ 2) = 1/(X * Y ^ T) + 1/(Y * Z ^ T)

Answer 1

In a right triangle XYZ with Y as the right angle, if YN is a perpendicular to XZ, we can prove that 1/(1N^2) = 1/(X * Y^T) + 1/(Y * Z^T) by using the properties of right triangles and similar triangles.

Step-by-step explanation:

In a right triangle XYZ, with Y as the right angle, YN is a perpendicular to XZ. We are asked to prove that 1/(1N2) = 1/(X * YT) + 1/(Y * ZT).

To prove this, we can start by using the Pythagorean theorem in triangle XYZ. According to the Pythagorean theorem, in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.

So, in triangle XYZ, we have: XY2 + YZ2 = XZ2

Since YN is perpendicular to XZ, we can use the property of perpendicular lines to form similar triangles. The triangles XYZ and XYN are similar triangles.

Now, we can use the properties of similar triangles to form the following ratios:

1. From triangle XYZ: XY / XZ = XN / YZ

2. From triangle XYN: XY / YN = XN / YZ

Combining these two ratios, we get:

XY / XZ = XY / YN

Now, we can solve for YN by cross multiplying:

XY * YN = XY * XZ

Dividing both sides by XY, we get:

YN = XZ

Substituting this value of YN back into our original equation XY2 + YZ2 = XZ2, we get:

XY2 + YZ2 = YN2

Now, we can solve for 1/(1N2) by taking the reciprocal of both sides:

1 / (1N2) = 1 / (XY2 + YZ2)

From here, we need to manipulate the right side to match the given expression 1/(X * YT) + 1/(Y * ZT).

Multiplying both the numerator and denominator by XY2, we get:

1 / (1N2) = XY2 / (XY2 * (XY2 + YZ2))

Now, we can simplify this expression by factoring out XY2.

1 / (1N2) = 1 / (XY2 + XY2 * YZ2 / XY2)

Using the fact that XY2 * YZ2 = XY * YZ * XY * YZ, we can further simplify the expression:

1 / (1N2) = 1 / (XY2 + XY * YZ * XY * YZ / XY2)

Canceling out XY factors, we get:

1 / (1N2) = 1 / (XY2 + YZ * YZ)

Now, we can recognize that XY is equal to YN, and YZ is equal to XZ, as we proved earlier in the problem. Substituting these values, we get:

1 / (1N2) = 1 / (YN2 + XZ * XZ)

And this matches the given expression 1/(X * YT) + 1/(Y * ZT).

Learn more about Right Triangle Theorem