Problems of similarity in triangles formed by an altitude from the right angle to the hypothenusa.

Angle-angle-angle (AAA) similarity
The angle-angle-angle (AAA) similarity test states that given two triangles that have corresponding angles that are congruent, then the triangles are similar. As we know the sum of the interior angles in a triangle is 180°, so if two corresponding are congruent, then the other ones should be as well.

Proportion
A statement that two ratios are equal.

Proportional
One of four numbers that form a true proportion.

Pythagorean theorem a2 + b2 = c2.
The Pythagorean theorem states that if you have a right triangle, then the square built on the hypotenuse is equal to the sum of the squares built on the other two sides.

Ratio
A quotient of 2 numbers.

Right triangle
A triangle that has a 90 degree angle.

Side-angle-side (SAS) similarity
The side-angle-side (SAS) similarity test states that given two triangles that have two pairs of sides that are proportional and the included angles are congruent, then the triangles should be similar.

Side-side-side (SSS) similarity
The side-side-side (SSS) similarity test states that for two triangles to be similar; all corresponding sides should be proportional.

Similar
Two polygons are similar polygons if corresponding angles have the same measure and corresponding sides are in proportion.

Similar triangles
Similar triangles are triangles which have the same shape but probably different size. Corresponding angles need to be congruent, and corresponding sides are in proportion.

Triangle
A polygon with three sides.

This great applet shows you, how you may draw three similar triangles, after you

draw the altitude from the hypotenuse to the right angle in a right triangle.

The triangles have been drawn with the same orientation. You may identify them

by the colors, the labels in the vertices, and the colors and measures of the side segments.

Drag point "B" to see how they update angles and side lengths to keep their

proportionality and similarity as shown in the provided extended proportions.

Similarity In A Right Triangle When An Altitude Is Drawn From The Hypotenuse To The Right Angle.

So far, we know how to solve similarity in triangles, but how about having a right triangle with the height from the hypotenuse? Can you see the embedded triangles in the figure? May you determine which sides are corresponding for them to setup proportions? Do you know why they are similar?

In the problem solving of these examples, you will explore the examples by separately drawing the embedded triangles and then relating the corresponding sides to build the proportions that will enable you to find the unknown segment lengths. It is a very visual approach, You will like it!