Another Proof of the Hypotenuse Leg Theorem

Given two triangles, if their hypotenuse are congruent, and one pair of their legs are congruent then the two triangles are congruent. In this post, we are going to prove this theorem.

In the figure below, ABC and DEF are right triangles with right angles at C and F, respectively.

hypotenuse leg 1

It is given that $latex \overline{AB} \cong \overline{DE}$ and $latex \overline{AC} \cong \overline{DF}$. We are going to prove that $latex \triangle ABC \cong \triangle DEF$.  Continue reading

Difference of 3-Digit Numbers with Reversed Digits

In the previous post, we proved a theorem that the sum of a 2-digit number with reversed digit is divisible by 11. In this post, we will learn another theorem about numbers with reversed digits. This time, we will explore the difference between two 3-digit numbers with reversed digits.

Let’s have the following examples.

231 – 132 = 99

512 – 215 = 297

741 – 147 = 594

543 – 345 = 198

Before continuing reading, try other numbers and see if you can find a pattern.

You have probably observed that all the differences of a 3-digit number and a number formed by reversing its digit is a multiple by 99. Now, is this true for all possible cases? You may want to prove this on your own before continuing below. The proof on the 2-digit number with reversed digits is would be a good hint.  Continue reading

Slope of Perpendicular Lines Theorem

If you have already learned about systems of linear equations, then you have probably discussed that the product of the slope of perpendicular lines is $latex -1$. The proof of this theorem comes from the fact that any point $latex (x,y)$ rotated 90 degrees at about the origin becomes $latex (-y,x)$. One example of this is shown below. The point $latex (3, 4)$, when rotated $latex 90$ degrees counterclockwise becomes $latex (-4,3)$.

With this fact, we prove this theorem.

Slope of Perpendicular Lines Theorem

If two lines with slopes $latex m_1$ and $latex m_2$ are perpendicular, then $latex m_1 m_2 = -1$.  Continue reading