4. **New Perpendicular Inside a Right Triangle:** - Spec…

But this perpendicular is not drawn to the midpoint of the hypotenuse.
This perpendicular splits the original hypotenuse, now the new base, into 2 parts: one part is 4.5cm and the other part is 1.5cm.
Right?

Yes, that's correct. When the perpendicular is drawn from the right-angle vertex to the new base (formerly the hypotenuse), it divides the base into two unequal segments: one measuring 4.5 cm and the other measuring 1.5 cm.

So if I want now to know the length of that perpendicular, what do you find?

To find the length of the perpendicular, you can use the concept of similar triangles that is often formed in right triangles when such perpendiculars are drawn. However, given the specific segment lengths you mentioned (4.5 cm and 1.5 cm), the perpendicular actually creates a smaller right triangle within the original triangle.

Since you have the segments of 4.5 cm and 1.5 cm, you can use the formula for the area of the triangles. The area of the triangle can be expressed in two different ways using the original right triangle configuration and the new configuration with the perpendicular.

Let \( p \) be the length of the perpendicular. The area of the larger triangle can be calculated with either base as:

Area (using the original hypotenuse) = (1/2) * 9 (original height) * 6 (original base),
Area (using the perpendicular and new base segments) = (1/2) * 6 (base) * p = (1/2) * (4.5 + 1.5) * p.

Thus:
9 * 3 = 6 * p,
27 = 6p,
p = 4.5

The length of the perpendicular is 4.5 cm.