In ΔABC, AB = BC and m∠ABC = 90°. D and E are the midpoints of AB and BC, respectively. If the length of CA is 9 units, the length of DE is _____ units and m∠CAB is _____ °.

In triangle ΔABC, since AB = BC and m∠ABC = 90°, it is an isosceles right triangle. Let’s denote the lengths of AB and BC as x.

Using the Pythagorean theorem, we know:

[

CA^2 = AB^2 + BC^2

]

[

9^2 = x^2 + x^2

]

[

81 = 2x^2

]

[

x^2 = frac{81}{2}

]

[

x = frac{9}{sqrt{2}} = frac{9sqrt{2}}{2} text{ (after rationalizing)}

]

Next, to find the length of DE, which is the segment connecting midpoints D and E, we use the midpoint theorem.

DE is parallel to AC and half its length:

[

DE = frac{1}{2} times CA = frac{1}{2} times 9 = 4.5

]

So, the length of DE is 4.5 units.

Now, to find m∠CAB, since it is a right triangle with equal legs, it follows that:

[

m∠CAB = 45°

]

Therefore, the answers are:

Length of DE is 4.5 units, and m∠CAB is 45 °.